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<a href="lagrangian__relaxation_8cc.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">// Copyright 2010-2021 Google LLC</span></div>
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<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span><span class="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div>
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<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="comment">// you may not use this file except in compliance with the License.</span></div>
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<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="comment">// You may obtain a copy of the License at</span></div>
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<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment">//</span></div>
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<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment">// http://www.apache.org/licenses/LICENSE-2.0</span></div>
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<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment">//</span></div>
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<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
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<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div>
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<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
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<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="comment">// See the License for the specific language governing permissions and</span></div>
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<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment">// limitations under the License.</span></div>
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<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span> </div>
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<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span><span class="comment">// Solves a constrained shortest path problem via Lagrangian Relaxation. The</span></div>
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<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="comment">// Lagrangian dual is solved with subgradient ascent.</span></div>
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<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span><span class="comment">//</span></div>
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<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="comment">// Problem data:</span></div>
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<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="comment">// * N: set of nodes.</span></div>
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<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span><span class="comment">// * A: set of arcs.</span></div>
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<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="comment">// * R: set of resources.</span></div>
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<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="comment">// * c_(i,j): cost of traversing arc (i,j) in A.</span></div>
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<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="comment">// * r_(i,j,k): resource k spent by traversing arc (i,j) in A, for all k in R.</span></div>
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<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="comment">// * b_i: flow balance at node i in N (+1 at the source, -1 at the sink, and 0</span></div>
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<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="comment">// otherwise).</span></div>
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<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span><span class="comment">// * r_max_k: availability of resource k for a path, for all k in R.</span></div>
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<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="comment">//</span></div>
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<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span><span class="comment">// Decision variables:</span></div>
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<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span><span class="comment">// * x_(i,j): flow through arc (i,j) in A.</span></div>
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<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span><span class="comment">//</span></div>
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<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span><span class="comment">// Formulation:</span></div>
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<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="comment">// Z = min sum(c_(i,j) * x_(i,j): (i,j) in A)</span></div>
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<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="comment">// s.t.</span></div>
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<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span><span class="comment">// sum(x_(i,j): (i,j) in A) - sum(x_(j,i): (j,i) in A) = b_i for all i in N,</span></div>
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<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span><span class="comment">// sum(r_(i,j,k) * x_(i,j): (i,j) in A) <= r_max_k for all k in R,</span></div>
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<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="comment">// x_(i,j) in {0,1} for all (i,j) in A.</span></div>
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<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="comment">//</span></div>
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<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span><span class="comment">// Upon dualizing a subset of the constraints (here we chose to relax some or</span></div>
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<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span><span class="comment">// all of the knapsack constraints), we obtaing a subproblem parameterized by</span></div>
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<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="comment">// dual variables mu (one per dualized constraint). We refer to this as the</span></div>
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<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span><span class="comment">// Lagrangian subproblem. Let R+ be the set of knapsack constraints that we</span></div>
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<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="comment">// keep, and R- the set of knapsack constraints that get dualized. The</span></div>
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<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="comment">// Lagrangian subproblem follows:</span></div>
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<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="comment">//</span></div>
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<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span><span class="comment">// z(mu) = min sum(</span></div>
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<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="comment">// (c_(i,j) - sum(mu_k * r_(i,j,k): k in R)) * x_(i,j): (i,j) in A)</span></div>
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<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span><span class="comment">// + sum(mu_k * r_max_k: k in R-)</span></div>
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<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span><span class="comment">// s.t.</span></div>
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<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span><span class="comment">// sum(x_(i,j): (i,j) in A) - sum(x_(j,i): (j,i) in A) = b_i for all i in N,</span></div>
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<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span><span class="comment">// sum(r_(i,j,k) * x_(i,j): (i,j) in A) <= r_max_k for all k in R+,</span></div>
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<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="comment">// x_(i,j) in {0,1} for all (i,j) in A.</span></div>
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<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span><span class="comment">//</span></div>
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<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span><span class="comment">// We seek to solve the Lagrangian dual, which is of the form:</span></div>
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span><span class="comment">// Z_D = max{ z(mu) : mu <=0 }. Concavity of z(mu) allows us to solve the</span></div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span><span class="comment">// Lagrangian dual with the iterates:</span></div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span><span class="comment">// mu_(t+1) = mu_t + step_size_t * grad_mu_t, where</span></div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span><span class="comment">// grad_mu_t = r_max - sum(t_(i,j) * x_(i,j)^t: (i,j) in A) is a subgradient of</span></div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span><span class="comment">// z(mu_t) and x^t is an optimal solution to the problem induced by z(mu_t).</span></div>
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<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span><span class="comment">//</span></div>
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<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span><span class="comment">// In general we have that Z_D <= Z. For convex problems, Z_D = Z. For MIPs,</span></div>
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<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span><span class="comment">// Z_LP <= Z_D <= Z, where Z_LP is the linear relaxation of the original</span></div>
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<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span><span class="comment">// problem.</span></div>
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<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span><span class="comment">//</span></div>
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<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span><span class="comment">// In this particular example, we use two resource constraints. Either</span></div>
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<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span><span class="comment">// constraint or both can be dualized via the flags `dualize_resource_1` and</span></div>
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<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span><span class="comment">// `dualize_resource_2`. If both constraints are dualized, we have that Z_LP =</span></div>
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<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span><span class="comment">// Z_D because the resulting Lagrangian subproblem can be solved as a linear</span></div>
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<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span><span class="comment">// program (i.e., the problem becomes a pure shortest path problem upon</span></div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span><span class="comment">// dualizing all the side constraints). When only one of the side constraints is</span></div>
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<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span><span class="comment">// dualized, we can have Z_LP <= Z_D because the resulting Lagrangian subproblem</span></div>
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<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span><span class="comment">// needs to be solved as an MIP. For the particular data used in this example,</span></div>
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<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span><span class="comment">// dualizing only the first resource constraint leads to Z_LP < Z_D, while</span></div>
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<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span><span class="comment">// dualizing only the second resource constraint leads to Z_LP = Z_D. In either</span></div>
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<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span><span class="comment">// case, solving the Lagrandual dual also provides an upper bound to Z.</span></div>
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<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span><span class="comment">//</span></div>
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<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span><span class="comment">// Usage: blaze build -c opt</span></div>
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<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span><span class="comment">// ortools/math_opt/examples:lagrangian_relaxation</span></div>
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<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span><span class="comment">// blaze-bin/ortools/math_opt/examples/lagrangian_relaxation</span></div>
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<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> </div>
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<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span><span class="preprocessor">#include <math.h></span></div>
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<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> </div>
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<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span><span class="preprocessor">#include <algorithm></span></div>
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<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span><span class="preprocessor">#include <iostream></span></div>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span><span class="preprocessor">#include <limits></span></div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span><span class="preprocessor">#include <memory></span></div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span><span class="preprocessor">#include <string></span></div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span><span class="preprocessor">#include <vector></span></div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> </div>
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<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span><span class="preprocessor">#include "absl/flags/flag.h"</span></div>
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<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span><span class="preprocessor">#include "absl/flags/parse.h"</span></div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span><span class="preprocessor">#include "absl/flags/usage.h"</span></div>
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<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span><span class="preprocessor">#include "absl/memory/memory.h"</span></div>
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<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span><span class="preprocessor">#include "absl/status/statusor.h"</span></div>
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<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span><span class="preprocessor">#include "absl/strings/str_format.h"</span></div>
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<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span><span class="preprocessor">#include "absl/strings/string_view.h"</span></div>
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<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span><span class="preprocessor">#include "<a class="code" href="container__logging_8h.html">ortools/base/container_logging.h</a>"</span></div>
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<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span><span class="preprocessor">#include "<a class="code" href="base_2logging_8h.html">ortools/base/logging.h</a>"</span></div>
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<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span><span class="preprocessor">#include "<a class="code" href="mathutil_8h.html">ortools/base/mathutil.h</a>"</span></div>
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<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span><span class="preprocessor">#include "<a class="code" href="math__opt_8h.html">ortools/math_opt/cpp/math_opt.h</a>"</span></div>
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<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> </div>
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<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a"> 100</a></span><a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a>(<span class="keywordtype">double</span>, step_size, 0.95,</div>
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<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> <span class="stringliteral">"Stepsize for gradient ascent, determined as step_size^t."</span>);</div>
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<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a38044bbf977275409524d07163ab3276"> 102</a></span><a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a>(<span class="keywordtype">int</span>, max_iterations, 1000,</div>
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<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> <span class="stringliteral">"Max number of iterations for gradient ascent."</span>);</div>
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<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a54d9cccfffed73a64555e2d29363fd57"> 104</a></span><a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a>(<span class="keywordtype">bool</span>, dualize_resource_1, <span class="keyword">true</span>,</div>
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<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <span class="stringliteral">"If true, the side constraint associated to resource 1 is dualized."</span>);</div>
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<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#af983917cd26cd3cc8e9de25abf287fe4"> 106</a></span><a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a>(<span class="keywordtype">bool</span>, dualize_resource_2, <span class="keyword">false</span>,</div>
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<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> <span class="stringliteral">"If true, the side constraint associated to resource 2 is dualized."</span>);</div>
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<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> </div>
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<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a69e141f2228adf5791514fd1632e6fd6"> 109</a></span><a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a>(<span class="keywordtype">bool</span>, lagrangian_output, <span class="keyword">false</span>,</div>
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<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> <span class="stringliteral">"If true, shows the iteration log of the subgradient ascent "</span></div>
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<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> <span class="stringliteral">"procedure use to solve the Lagrangian problem"</span>);</div>
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<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> </div>
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<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a6ffb9a02546adae760787b2366ae6e41"> 113</a></span><span class="keyword">constexpr</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="lagrangian__relaxation_8cc.html#a6ffb9a02546adae760787b2366ae6e41">kZeroTol</a> = 1.0e-8;</div>
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<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> </div>
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<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span><span class="keyword">namespace </span>{</div>
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<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span>using ::operations_research::MathUtil;</div>
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<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span>using ::operations_research::math_opt::LinearExpression;</div>
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<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span>using ::operations_research::math_opt::Model;</div>
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<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span>using ::operations_research::math_opt::SolveArguments;</div>
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<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span>using ::operations_research::math_opt::SolveResult;</div>
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<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span><a class="code hl_enumeration" href="namespaceoperations__research_1_1math__opt.html#a8b06041d7c1fb05f379714f4312306ec">using ::operations_research::math_opt::SolverType</a>;</div>
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<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span><a class="code hl_enumeration" href="namespaceoperations__research_1_1math__opt.html#ad02e69a0531469b463df907c7b2ad194">using ::operations_research::math_opt::TerminationReason</a>;</div>
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<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span>using ::operations_research::math_opt::Variable;</div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span><a class="code hl_typedef" href="namespaceoperations__research_1_1math__opt.html#a252c66967569c5ab1db4b4d356707fb1">using ::operations_research::math_opt::VariableMap</a>;</div>
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<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> </div>
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<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span><span class="keyword">struct </span><a class="code hl_typedef" href="namespaceoperations__research.html#a7deeae530369e107f9d91c1a189f451f">Arc</a> {</div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> <span class="keywordtype">int</span> i;</div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> <span class="keywordtype">int</span> j;</div>
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<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>;</div>
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<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> <span class="keywordtype">double</span> resource_1;</div>
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<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> <span class="keywordtype">double</span> resource_2;</div>
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<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span>};</div>
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<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> </div>
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<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span><span class="keyword">struct </span><a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> {</div>
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<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> <span class="keywordtype">int</span> num_nodes;</div>
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<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> std::vector<Arc> arcs;</div>
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<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> <span class="keywordtype">int</span> source;</div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> <span class="keywordtype">int</span> sink;</div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span>};</div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> </div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span><span class="keyword">struct </span>FlowModel {</div>
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<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> FlowModel() : <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>(std::make_unique<Model>(<span class="stringliteral">"LagrangianProblem"</span>)) {}</div>
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<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> std::unique_ptr<Model> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>;</div>
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<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> LinearExpression <a class="code hl_variable" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>;</div>
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<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> LinearExpression resource_1;</div>
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<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> LinearExpression resource_2;</div>
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<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> std::vector<Variable> flow_vars;</div>
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<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span>};</div>
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<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> </div>
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<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span><span class="comment">// Populates `model` with variables and constraints of a shortest path problem.</span></div>
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<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span>FlowModel CreateShortestPathModel(<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> graph) {</div>
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<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> FlowModel flow_model;</div>
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<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> Model& <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a> = *flow_model.model;</div>
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<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7deeae530369e107f9d91c1a189f451f">Arc</a>& arc : graph.arcs) {</div>
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<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> Variable <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a> = <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddContinuousVariable(</div>
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<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="comment">/*lower_bound=*/</span>0, <span class="comment">/*upper_bound=*/</span>1,</div>
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="comment">/*name=*/</span>absl::StrFormat(<span class="stringliteral">"x_%d_%d"</span>, arc.i, arc.j));</div>
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<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> flow_model.cost += arc.cost * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> flow_model.resource_1 += arc.resource_1 * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> flow_model.resource_2 += arc.resource_2 * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> flow_model.flow_vars.push_back(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>);</div>
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<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> }</div>
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<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> </div>
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<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> <span class="comment">// Flow balance constraints</span></div>
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<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> std::vector<LinearExpression> out_flow(graph.num_nodes);</div>
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<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> std::vector<LinearExpression> in_flow(graph.num_nodes);</div>
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<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> arc_index = 0; arc_index < graph.arcs.size(); ++arc_index) {</div>
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<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> out_flow[graph.arcs[arc_index].i] += flow_model.flow_vars[arc_index];</div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> in_flow[graph.arcs[arc_index].j] += flow_model.flow_vars[arc_index];</div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> }</div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> node_index = 0; node_index < graph.num_nodes; ++node_index) {</div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> <span class="keywordtype">int</span> rhs = node_index == graph.source ? 1</div>
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<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> : node_index == graph.sink ? -1</div>
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<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> : 0;</div>
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<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(out_flow[node_index] - in_flow[node_index] ==</div>
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<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> rhs);</div>
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<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> }</div>
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<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> </div>
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<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> <span class="keywordflow">return</span> flow_model;</div>
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<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span>}</div>
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<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> </div>
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<div class="line"><a id="l00182" name="l00182"></a><span class="lineno"> 182</span><a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> CreateSampleNetwork() {</div>
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<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"> 183</span> std::vector<Arc> arcs;</div>
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<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span> arcs.push_back(</div>
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<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> {.i = 0, .j = 1, .cost = 12, .resource_1 = 1, .resource_2 = 1});</div>
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<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span> arcs.push_back(</div>
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<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span> {.i = 0, .j = 2, .cost = 3, .resource_1 = 2.5, .resource_2 = 1});</div>
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<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> arcs.push_back(</div>
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<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"> 189</span> {.i = 1, .j = 3, .cost = 5, .resource_1 = 1, .resource_2 = 1.5});</div>
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<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> arcs.push_back(</div>
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<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span> {.i = 1, .j = 4, .cost = 5, .resource_1 = 2.5, .resource_2 = 1});</div>
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<div class="line"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> arcs.push_back(</div>
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<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> {.i = 2, .j = 1, .cost = 7, .resource_1 = 2.5, .resource_2 = 1});</div>
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<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> arcs.push_back(</div>
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<div class="line"><a id="l00195" name="l00195"></a><span class="lineno"> 195</span> {.i = 2, .j = 3, .cost = 5, .resource_1 = 7, .resource_2 = 2.5});</div>
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<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span> arcs.push_back(</div>
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<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span> {.i = 2, .j = 4, .cost = 1, .resource_1 = 6.5, .resource_2 = 1});</div>
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<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span> arcs.push_back(</div>
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<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> {.i = 3, .j = 5, .cost = 6, .resource_1 = 1, .resource_2 = 2.0});</div>
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<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> arcs.push_back(</div>
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<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> {.i = 4, .j = 3, .cost = 3, .resource_1 = 1, .resource_2 = 0.5});</div>
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<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span> arcs.push_back(</div>
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<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> {.i = 4, .j = 5, .cost = 5, .resource_1 = 2.5, .resource_2 = 1});</div>
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<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> graph = {.num_nodes = 6, .arcs = arcs, .source = 0, .sink = 5};</div>
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<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span> </div>
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<div class="line"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> <span class="keywordflow">return</span> graph;</div>
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<div class="line"><a id="l00207" name="l00207"></a><span class="lineno"> 207</span>}</div>
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<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> </div>
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<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span><span class="comment">// Solves the constrained shortest path as an MIP.</span></div>
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<div class="line"><a id="l00210" name="l00210"></a><span class="lineno"> 210</span>FlowModel SolveMip(<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> graph, <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_1,</div>
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<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_2) {</div>
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<div class="line"><a id="l00212" name="l00212"></a><span class="lineno"> 212</span> FlowModel flow_model;</div>
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<div class="line"><a id="l00213" name="l00213"></a><span class="lineno"> 213</span> Model& <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a> = *flow_model.model;</div>
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<div class="line"><a id="l00214" name="l00214"></a><span class="lineno"> 214</span> <span class="keywordflow">for</span> (<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceoperations__research.html#a7deeae530369e107f9d91c1a189f451f">Arc</a>& arc : graph.arcs) {</div>
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<div class="line"><a id="l00215" name="l00215"></a><span class="lineno"> 215</span> Variable <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a> = <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddBinaryVariable(</div>
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<div class="line"><a id="l00216" name="l00216"></a><span class="lineno"> 216</span> <span class="comment">/*name=*/</span>absl::StrFormat(<span class="stringliteral">"x_%d_%d"</span>, arc.i, arc.j));</div>
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<div class="line"><a id="l00217" name="l00217"></a><span class="lineno"> 217</span> flow_model.cost += arc.cost * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00218" name="l00218"></a><span class="lineno"> 218</span> flow_model.resource_1 += +arc.resource_1 * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00219" name="l00219"></a><span class="lineno"> 219</span> flow_model.resource_2 += arc.resource_2 * <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>;</div>
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<div class="line"><a id="l00220" name="l00220"></a><span class="lineno"> 220</span> flow_model.flow_vars.push_back(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>);</div>
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<div class="line"><a id="l00221" name="l00221"></a><span class="lineno"> 221</span> }</div>
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<div class="line"><a id="l00222" name="l00222"></a><span class="lineno"> 222</span> </div>
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<div class="line"><a id="l00223" name="l00223"></a><span class="lineno"> 223</span> <span class="comment">// Flow balance constraints</span></div>
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<div class="line"><a id="l00224" name="l00224"></a><span class="lineno"> 224</span> std::vector<LinearExpression> out_flow(graph.num_nodes);</div>
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<div class="line"><a id="l00225" name="l00225"></a><span class="lineno"> 225</span> std::vector<LinearExpression> in_flow(graph.num_nodes);</div>
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<div class="line"><a id="l00226" name="l00226"></a><span class="lineno"> 226</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> arc_index = 0; arc_index < graph.arcs.size(); ++arc_index) {</div>
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<div class="line"><a id="l00227" name="l00227"></a><span class="lineno"> 227</span> out_flow[graph.arcs[arc_index].i] += flow_model.flow_vars[arc_index];</div>
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<div class="line"><a id="l00228" name="l00228"></a><span class="lineno"> 228</span> in_flow[graph.arcs[arc_index].j] += flow_model.flow_vars[arc_index];</div>
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<div class="line"><a id="l00229" name="l00229"></a><span class="lineno"> 229</span> }</div>
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<div class="line"><a id="l00230" name="l00230"></a><span class="lineno"> 230</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> node_index = 0; node_index < graph.num_nodes; ++node_index) {</div>
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<div class="line"><a id="l00231" name="l00231"></a><span class="lineno"> 231</span> <span class="keywordtype">int</span> rhs = node_index == graph.source ? 1</div>
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<div class="line"><a id="l00232" name="l00232"></a><span class="lineno"> 232</span> : node_index == graph.sink ? -1</div>
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<div class="line"><a id="l00233" name="l00233"></a><span class="lineno"> 233</span> : 0;</div>
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<div class="line"><a id="l00234" name="l00234"></a><span class="lineno"> 234</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(out_flow[node_index] - in_flow[node_index] ==</div>
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<div class="line"><a id="l00235" name="l00235"></a><span class="lineno"> 235</span> rhs);</div>
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<div class="line"><a id="l00236" name="l00236"></a><span class="lineno"> 236</span> }</div>
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<div class="line"><a id="l00237" name="l00237"></a><span class="lineno"> 237</span> </div>
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<div class="line"><a id="l00238" name="l00238"></a><span class="lineno"> 238</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(flow_model.resource_1 <= max_resource_1,</div>
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<div class="line"><a id="l00239" name="l00239"></a><span class="lineno"> 239</span> <span class="stringliteral">"resource_ctr_1"</span>);</div>
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<div class="line"><a id="l00240" name="l00240"></a><span class="lineno"> 240</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(flow_model.resource_2 <= max_resource_2,</div>
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<div class="line"><a id="l00241" name="l00241"></a><span class="lineno"> 241</span> <span class="stringliteral">"resource_ctr_2"</span>);</div>
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<div class="line"><a id="l00242" name="l00242"></a><span class="lineno"> 242</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.Minimize(flow_model.cost);</div>
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<div class="line"><a id="l00243" name="l00243"></a><span class="lineno"> 243</span> <span class="keyword">const</span> SolveResult result = <a class="code hl_function" href="namespaceoperations__research_1_1math__opt.html#a5767ee23f380e72488d3c7ebf2d742b1">Solve</a>(<a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>, SolverType::kGscip).value();</div>
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<div class="line"><a id="l00244" name="l00244"></a><span class="lineno"> 244</span> <span class="keyword">const</span> VariableMap<double>& variable_values = result.variable_values();</div>
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<div class="line"><a id="l00245" name="l00245"></a><span class="lineno"> 245</span> std::cout << <span class="stringliteral">"MIP Solution with 2 side constraints"</span> << std::endl;</div>
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<div class="line"><a id="l00246" name="l00246"></a><span class="lineno"> 246</span> std::cout << absl::StrFormat(<span class="stringliteral">"MIP objective value: %6.3f"</span>,</div>
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<div class="line"><a id="l00247" name="l00247"></a><span class="lineno"> 247</span> result.objective_value())</div>
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<div class="line"><a id="l00248" name="l00248"></a><span class="lineno"> 248</span> << std::endl;</div>
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<div class="line"><a id="l00249" name="l00249"></a><span class="lineno"> 249</span> std::cout << <span class="stringliteral">"Resource 1: "</span> << flow_model.resource_1.Evaluate(variable_values)</div>
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<div class="line"><a id="l00250" name="l00250"></a><span class="lineno"> 250</span> << std::endl;</div>
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<div class="line"><a id="l00251" name="l00251"></a><span class="lineno"> 251</span> std::cout << <span class="stringliteral">"Resource 2: "</span> << flow_model.resource_2.Evaluate(variable_values)</div>
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<div class="line"><a id="l00252" name="l00252"></a><span class="lineno"> 252</span> << std::endl;</div>
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<div class="line"><a id="l00253" name="l00253"></a><span class="lineno"> 253</span> std::cout << <span class="stringliteral">"========================================"</span> << std::endl;</div>
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<div class="line"><a id="l00254" name="l00254"></a><span class="lineno"> 254</span> <span class="keywordflow">return</span> flow_model;</div>
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<div class="line"><a id="l00255" name="l00255"></a><span class="lineno"> 255</span>}</div>
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<div class="line"><a id="l00256" name="l00256"></a><span class="lineno"> 256</span> </div>
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<div class="line"><a id="l00257" name="l00257"></a><span class="lineno"> 257</span><span class="comment">// Solves the linear relaxation of a constrained shortest path problem</span></div>
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<div class="line"><a id="l00258" name="l00258"></a><span class="lineno"> 258</span><span class="comment">// formulated as an MIP.</span></div>
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<div class="line"><a id="l00259" name="l00259"></a><span class="lineno"> 259</span><span class="keywordtype">void</span> SolveLinearRelaxation(FlowModel& flow_model, <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph,</div>
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<div class="line"><a id="l00260" name="l00260"></a><span class="lineno"> 260</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_1,</div>
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<div class="line"><a id="l00261" name="l00261"></a><span class="lineno"> 261</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_2) {</div>
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<div class="line"><a id="l00262" name="l00262"></a><span class="lineno"> 262</span> Model& <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a> = *flow_model.model;</div>
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<div class="line"><a id="l00263" name="l00263"></a><span class="lineno"> 263</span> <span class="keyword">const</span> SolveResult result = <a class="code hl_function" href="namespaceoperations__research_1_1math__opt.html#a5767ee23f380e72488d3c7ebf2d742b1">Solve</a>(<a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>, SolverType::kGscip).value();</div>
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<div class="line"><a id="l00264" name="l00264"></a><span class="lineno"> 264</span> <span class="keyword">const</span> VariableMap<double>& variable_values = result.variable_values();</div>
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<div class="line"><a id="l00265" name="l00265"></a><span class="lineno"> 265</span> std::cout << <span class="stringliteral">"LP relaxation with 2 side constraints"</span> << std::endl;</div>
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<div class="line"><a id="l00266" name="l00266"></a><span class="lineno"> 266</span> std::cout << absl::StrFormat(<span class="stringliteral">"LP objective value: %6.3f"</span>,</div>
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<div class="line"><a id="l00267" name="l00267"></a><span class="lineno"> 267</span> result.objective_value())</div>
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<div class="line"><a id="l00268" name="l00268"></a><span class="lineno"> 268</span> << std::endl;</div>
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<div class="line"><a id="l00269" name="l00269"></a><span class="lineno"> 269</span> std::cout << <span class="stringliteral">"Resource 1: "</span> << flow_model.resource_1.Evaluate(variable_values)</div>
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<div class="line"><a id="l00270" name="l00270"></a><span class="lineno"> 270</span> << std::endl;</div>
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<div class="line"><a id="l00271" name="l00271"></a><span class="lineno"> 271</span> std::cout << <span class="stringliteral">"Resource 2: "</span> << flow_model.resource_2.Evaluate(variable_values)</div>
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<div class="line"><a id="l00272" name="l00272"></a><span class="lineno"> 272</span> << std::endl;</div>
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<div class="line"><a id="l00273" name="l00273"></a><span class="lineno"> 273</span> std::cout << <span class="stringliteral">"========================================"</span> << std::endl;</div>
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<div class="line"><a id="l00274" name="l00274"></a><span class="lineno"> 274</span>}</div>
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<div class="line"><a id="l00275" name="l00275"></a><span class="lineno"> 275</span> </div>
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<div class="line"><a id="l00276" name="l00276"></a><span class="lineno"> 276</span><span class="keywordtype">void</span> SolveLagrangianRelaxation(<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> graph, <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_1,</div>
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<div class="line"><a id="l00277" name="l00277"></a><span class="lineno"> 277</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_2) {</div>
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<div class="line"><a id="l00278" name="l00278"></a><span class="lineno"> 278</span> <span class="comment">// Model, variables, and linear expressions.</span></div>
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<div class="line"><a id="l00279" name="l00279"></a><span class="lineno"> 279</span> FlowModel flow_model = CreateShortestPathModel(graph);</div>
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<div class="line"><a id="l00280" name="l00280"></a><span class="lineno"> 280</span> Model& <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a> = *flow_model.model;</div>
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<div class="line"><a id="l00281" name="l00281"></a><span class="lineno"> 281</span> LinearExpression& <a class="code hl_variable" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a> = flow_model.cost;</div>
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<div class="line"><a id="l00282" name="l00282"></a><span class="lineno"> 282</span> LinearExpression& resource_1 = flow_model.resource_1;</div>
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<div class="line"><a id="l00283" name="l00283"></a><span class="lineno"> 283</span> LinearExpression& resource_2 = flow_model.resource_2;</div>
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<div class="line"><a id="l00284" name="l00284"></a><span class="lineno"> 284</span> </div>
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<div class="line"><a id="l00285" name="l00285"></a><span class="lineno"> 285</span> <span class="comment">// Dualized constraints and dual variable iterates.</span></div>
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<div class="line"><a id="l00286" name="l00286"></a><span class="lineno"> 286</span> std::vector<double> mu;</div>
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<div class="line"><a id="l00287" name="l00287"></a><span class="lineno"> 287</span> std::vector<LinearExpression> grad_mu;</div>
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<div class="line"><a id="l00288" name="l00288"></a><span class="lineno"> 288</span> <span class="keyword">const</span> <span class="keywordtype">bool</span> dualized_resource_1 = absl::GetFlag(FLAGS_dualize_resource_1);</div>
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<div class="line"><a id="l00289" name="l00289"></a><span class="lineno"> 289</span> <span class="keyword">const</span> <span class="keywordtype">bool</span> dualized_resource_2 = absl::GetFlag(FLAGS_dualize_resource_2);</div>
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<div class="line"><a id="l00290" name="l00290"></a><span class="lineno"> 290</span> <span class="keyword">const</span> <span class="keywordtype">bool</span> lagrangian_output = absl::GetFlag(FLAGS_lagrangian_output);</div>
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<div class="line"><a id="l00291" name="l00291"></a><span class="lineno"> 291</span> <a class="code hl_define" href="base_2logging_8h.html#a3e1cfef60e774a81f30eaddf26a3a274">CHECK</a>(dualized_resource_1 || dualized_resource_2)</div>
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<div class="line"><a id="l00292" name="l00292"></a><span class="lineno"> 292</span> << <span class="stringliteral">"At least one of the side constraints should be dualized."</span>;</div>
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<div class="line"><a id="l00293" name="l00293"></a><span class="lineno"> 293</span> </div>
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<div class="line"><a id="l00294" name="l00294"></a><span class="lineno"> 294</span> <span class="comment">// Modify the lagrangian problem according to the constraints that are</span></div>
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<div class="line"><a id="l00295" name="l00295"></a><span class="lineno"> 295</span> <span class="comment">// dualized. We use a initial dual value different from zero to prioritize</span></div>
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<div class="line"><a id="l00296" name="l00296"></a><span class="lineno"> 296</span> <span class="comment">// finding a feasible solution.</span></div>
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<div class="line"><a id="l00297" name="l00297"></a><span class="lineno"> 297</span> <span class="keyword">const</span> <span class="keywordtype">double</span> initial_dual_value = -10;</div>
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<div class="line"><a id="l00298" name="l00298"></a><span class="lineno"> 298</span> <span class="keywordflow">if</span> (dualized_resource_1 && !dualized_resource_2) {</div>
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<div class="line"><a id="l00299" name="l00299"></a><span class="lineno"> 299</span> mu.push_back(initial_dual_value);</div>
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<div class="line"><a id="l00300" name="l00300"></a><span class="lineno"> 300</span> grad_mu.push_back(max_resource_1 - resource_1);</div>
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<div class="line"><a id="l00301" name="l00301"></a><span class="lineno"> 301</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(resource_2 <= max_resource_2);</div>
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<div class="line"><a id="l00302" name="l00302"></a><span class="lineno"> 302</span> <span class="keywordflow">for</span> (Variable& <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a> : flow_model.flow_vars) {</div>
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<div class="line"><a id="l00303" name="l00303"></a><span class="lineno"> 303</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.set_integer(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>);</div>
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<div class="line"><a id="l00304" name="l00304"></a><span class="lineno"> 304</span> }</div>
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<div class="line"><a id="l00305" name="l00305"></a><span class="lineno"> 305</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (!dualized_resource_1 && dualized_resource_2) {</div>
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<div class="line"><a id="l00306" name="l00306"></a><span class="lineno"> 306</span> mu.push_back(initial_dual_value);</div>
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<div class="line"><a id="l00307" name="l00307"></a><span class="lineno"> 307</span> grad_mu.push_back(max_resource_2 - resource_2);</div>
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<div class="line"><a id="l00308" name="l00308"></a><span class="lineno"> 308</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.AddLinearConstraint(resource_1 <= max_resource_1);</div>
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<div class="line"><a id="l00309" name="l00309"></a><span class="lineno"> 309</span> <span class="keywordflow">for</span> (Variable& <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a> : flow_model.flow_vars) {</div>
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<div class="line"><a id="l00310" name="l00310"></a><span class="lineno"> 310</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.set_integer(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>);</div>
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<div class="line"><a id="l00311" name="l00311"></a><span class="lineno"> 311</span> }</div>
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<div class="line"><a id="l00312" name="l00312"></a><span class="lineno"> 312</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><a id="l00313" name="l00313"></a><span class="lineno"> 313</span> mu.push_back(initial_dual_value);</div>
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<div class="line"><a id="l00314" name="l00314"></a><span class="lineno"> 314</span> mu.push_back(initial_dual_value);</div>
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<div class="line"><a id="l00315" name="l00315"></a><span class="lineno"> 315</span> grad_mu.push_back(max_resource_1 - resource_1);</div>
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<div class="line"><a id="l00316" name="l00316"></a><span class="lineno"> 316</span> grad_mu.push_back(max_resource_2 - resource_2);</div>
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<div class="line"><a id="l00317" name="l00317"></a><span class="lineno"> 317</span> }</div>
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<div class="line"><a id="l00318" name="l00318"></a><span class="lineno"> 318</span> </div>
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<div class="line"><a id="l00319" name="l00319"></a><span class="lineno"> 319</span> <span class="comment">// Gradient ascent setup</span></div>
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<div class="line"><a id="l00320" name="l00320"></a><span class="lineno"> 320</span> <span class="keywordtype">bool</span> termination = <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00321" name="l00321"></a><span class="lineno"> 321</span> <span class="keywordtype">int</span> iterations = 1;</div>
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<div class="line"><a id="l00322" name="l00322"></a><span class="lineno"> 322</span> <span class="keyword">const</span> <span class="keywordtype">double</span> step_size = absl::GetFlag(FLAGS_step_size);</div>
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<div class="line"><a id="l00323" name="l00323"></a><span class="lineno"> 323</span> <a class="code hl_define" href="base_2logging_8h.html#a7e03ec13560fa94a8fea569960d7efc6">CHECK_GT</a>(step_size, 0) << <span class="stringliteral">"step_size must be strictly positive"</span>;</div>
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<div class="line"><a id="l00324" name="l00324"></a><span class="lineno"> 324</span> <a class="code hl_define" href="base_2logging_8h.html#a4bd2e815ca2f702a4b6aa744b1ff3b82">CHECK_LT</a>(step_size, 1) << <span class="stringliteral">"step_size must be strictly less than 1"</span>;</div>
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<div class="line"><a id="l00325" name="l00325"></a><span class="lineno"> 325</span> <span class="keyword">const</span> <span class="keywordtype">int</span> max_iterations = absl::GetFlag(FLAGS_max_iterations);</div>
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<div class="line"><a id="l00326" name="l00326"></a><span class="lineno"> 326</span> <a class="code hl_define" href="base_2logging_8h.html#a7e03ec13560fa94a8fea569960d7efc6">CHECK_GT</a>(max_iterations, 0)</div>
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<div class="line"><a id="l00327" name="l00327"></a><span class="lineno"> 327</span> << <span class="stringliteral">"Number of iterations must be strictly positive."</span>;</div>
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<div class="line"><a id="l00328" name="l00328"></a><span class="lineno"> 328</span> </div>
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<div class="line"><a id="l00329" name="l00329"></a><span class="lineno"> 329</span> <span class="comment">// Upper and lower bounds on the full problem.</span></div>
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<div class="line"><a id="l00330" name="l00330"></a><span class="lineno"> 330</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a> = std::numeric_limits<double>::infinity();</div>
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<div class="line"><a id="l00331" name="l00331"></a><span class="lineno"> 331</span> <span class="keywordtype">double</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a> = -std::numeric_limits<double>::infinity();</div>
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<div class="line"><a id="l00332" name="l00332"></a><span class="lineno"> 332</span> <span class="keywordtype">double</span> best_solution_resource_1 = 0;</div>
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<div class="line"><a id="l00333" name="l00333"></a><span class="lineno"> 333</span> <span class="keywordtype">double</span> best_solution_resource_2 = 0;</div>
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<div class="line"><a id="l00334" name="l00334"></a><span class="lineno"> 334</span> </div>
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<div class="line"><a id="l00335" name="l00335"></a><span class="lineno"> 335</span> <span class="keywordflow">if</span> (lagrangian_output) {</div>
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<div class="line"><a id="l00336" name="l00336"></a><span class="lineno"> 336</span> std::cout << <span class="stringliteral">"Starting gradient ascent..."</span> << std::endl;</div>
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<div class="line"><a id="l00337" name="l00337"></a><span class="lineno"> 337</span> std::cout << absl::StrFormat(<span class="stringliteral">"%4s %6s %6s %9s %10s %10s"</span>, <span class="stringliteral">"Iter"</span>, <span class="stringliteral">"LB"</span>,</div>
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<div class="line"><a id="l00338" name="l00338"></a><span class="lineno"> 338</span> <span class="stringliteral">"UB"</span>, <span class="stringliteral">"Step size"</span>, <span class="stringliteral">"mu_t"</span>, <span class="stringliteral">"grad_mu_t"</span>)</div>
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<div class="line"><a id="l00339" name="l00339"></a><span class="lineno"> 339</span> << std::endl;</div>
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<div class="line"><a id="l00340" name="l00340"></a><span class="lineno"> 340</span> }</div>
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<div class="line"><a id="l00341" name="l00341"></a><span class="lineno"> 341</span> </div>
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<div class="line"><a id="l00342" name="l00342"></a><span class="lineno"> 342</span> <span class="keywordflow">while</span> (!termination) {</div>
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<div class="line"><a id="l00343" name="l00343"></a><span class="lineno"> 343</span> LinearExpression lagrangian_function;</div>
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<div class="line"><a id="l00344" name="l00344"></a><span class="lineno"> 344</span> lagrangian_function += <a class="code hl_variable" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>;</div>
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<div class="line"><a id="l00345" name="l00345"></a><span class="lineno"> 345</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; k < mu.size(); ++k) {</div>
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<div class="line"><a id="l00346" name="l00346"></a><span class="lineno"> 346</span> lagrangian_function += mu[k] * grad_mu[k];</div>
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<div class="line"><a id="l00347" name="l00347"></a><span class="lineno"> 347</span> }</div>
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<div class="line"><a id="l00348" name="l00348"></a><span class="lineno"> 348</span> <a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>.Minimize(lagrangian_function);</div>
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<div class="line"><a id="l00349" name="l00349"></a><span class="lineno"> 349</span> SolveResult result = <a class="code hl_function" href="namespaceoperations__research_1_1math__opt.html#a5767ee23f380e72488d3c7ebf2d742b1">Solve</a>(<a class="code hl_variable" href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a>, SolverType::kGscip).value();</div>
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<div class="line"><a id="l00350" name="l00350"></a><span class="lineno"> 350</span> <a class="code hl_define" href="base_2logging_8h.html#a7c0ce053b28d53aa4eaf3eb7fb71663b">CHECK_EQ</a>(result.termination.reason, TerminationReason::kOptimal)</div>
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<div class="line"><a id="l00351" name="l00351"></a><span class="lineno"> 351</span> << result.termination;</div>
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<div class="line"><a id="l00352" name="l00352"></a><span class="lineno"> 352</span> <span class="keyword">const</span> VariableMap<double>& vars_val = result.variable_values();</div>
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<div class="line"><a id="l00353" name="l00353"></a><span class="lineno"> 353</span> <span class="keywordtype">bool</span> feasible = <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00354" name="l00354"></a><span class="lineno"> 354</span> </div>
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<div class="line"><a id="l00355" name="l00355"></a><span class="lineno"> 355</span> <span class="comment">// Iterate update. Takes a step in the direction of the gradient (since the</span></div>
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<div class="line"><a id="l00356" name="l00356"></a><span class="lineno"> 356</span> <span class="comment">// Lagrangian dual is a max problem), and projects onto {mu: mu <=0} to</span></div>
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<div class="line"><a id="l00357" name="l00357"></a><span class="lineno"> 357</span> <span class="comment">// satisfy the sign of the dual variable. In general, convergence to an</span></div>
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<div class="line"><a id="l00358" name="l00358"></a><span class="lineno"> 358</span> <span class="comment">// optimal solution requires diminishing step sizes satisfying:</span></div>
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<div class="line"><a id="l00359" name="l00359"></a><span class="lineno"> 359</span> <span class="comment">// * sum(step_size_t: t=1...) = infinity and,</span></div>
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<div class="line"><a id="l00360" name="l00360"></a><span class="lineno"> 360</span> <span class="comment">// * sum((step_size_t)^2: t=1...) < infinity</span></div>
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<div class="line"><a id="l00361" name="l00361"></a><span class="lineno"> 361</span> <span class="comment">// See details in Prop 3.2.6 Bertsekas 2015, Convex Optimization Algorithms.</span></div>
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<div class="line"><a id="l00362" name="l00362"></a><span class="lineno"> 362</span> <span class="comment">// Here we use step_size_t = step_size^t which does NOT satisfy the</span></div>
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<div class="line"><a id="l00363" name="l00363"></a><span class="lineno"> 363</span> <span class="comment">// first condition, but is good enough for the purpose of this example.</span></div>
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<div class="line"><a id="l00364" name="l00364"></a><span class="lineno"> 364</span> std::vector<double> grad_mu_vals;</div>
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<div class="line"><a id="l00365" name="l00365"></a><span class="lineno"> 365</span> <span class="keyword">const</span> <span class="keywordtype">double</span> step_size_t = MathUtil::IPow(step_size, iterations);</div>
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<div class="line"><a id="l00366" name="l00366"></a><span class="lineno"> 366</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; k < mu.size(); ++k) {</div>
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<div class="line"><a id="l00367" name="l00367"></a><span class="lineno"> 367</span> <span class="comment">// Evaluate resource k and evaluate the gradient of z(mu).</span></div>
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<div class="line"><a id="l00368" name="l00368"></a><span class="lineno"> 368</span> <span class="keyword">const</span> <span class="keywordtype">double</span> grad_mu_val = grad_mu[k].Evaluate(vars_val);</div>
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<div class="line"><a id="l00369" name="l00369"></a><span class="lineno"> 369</span> grad_mu_vals.push_back(grad_mu_val);</div>
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<div class="line"><a id="l00370" name="l00370"></a><span class="lineno"> 370</span> mu[k] = <a class="code hl_variable" href="alldiff__cst_8cc.html#ad10edae0a852d72fb76afb1c77735045">std::min</a>(0.0, mu[k] + step_size_t * grad_mu_val);</div>
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<div class="line"><a id="l00371" name="l00371"></a><span class="lineno"> 371</span> <span class="keywordflow">if</span> (grad_mu_val < 0) {</div>
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<div class="line"><a id="l00372" name="l00372"></a><span class="lineno"> 372</span> feasible = <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00373" name="l00373"></a><span class="lineno"> 373</span> }</div>
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<div class="line"><a id="l00374" name="l00374"></a><span class="lineno"> 374</span> }</div>
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<div class="line"><a id="l00375" name="l00375"></a><span class="lineno"> 375</span> <span class="comment">// Bounds update</span></div>
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<div class="line"><a id="l00376" name="l00376"></a><span class="lineno"> 376</span> <span class="keyword">const</span> <span class="keywordtype">double</span> path_cost = <a class="code hl_variable" href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a>.Evaluate(vars_val);</div>
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<div class="line"><a id="l00377" name="l00377"></a><span class="lineno"> 377</span> <span class="keywordflow">if</span> (feasible && path_cost < <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a>) {</div>
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<div class="line"><a id="l00378" name="l00378"></a><span class="lineno"> 378</span> best_solution_resource_1 = resource_1.Evaluate(vars_val);</div>
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<div class="line"><a id="l00379" name="l00379"></a><span class="lineno"> 379</span> best_solution_resource_2 = resource_2.Evaluate(vars_val);</div>
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<div class="line"><a id="l00380" name="l00380"></a><span class="lineno"> 380</span> <span class="keywordflow">if</span> (lagrangian_output) {</div>
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<div class="line"><a id="l00381" name="l00381"></a><span class="lineno"> 381</span> std::cout << <span class="stringliteral">"Feasible solution with"</span></div>
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<div class="line"><a id="l00382" name="l00382"></a><span class="lineno"> 382</span> << absl::StrFormat(</div>
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<div class="line"><a id="l00383" name="l00383"></a><span class="lineno"> 383</span> <span class="stringliteral">"cost=%4.2f, resource_1=%4.2f, and resource_2=%4.2f. "</span>,</div>
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<div class="line"><a id="l00384" name="l00384"></a><span class="lineno"> 384</span> path_cost, best_solution_resource_1,</div>
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<div class="line"><a id="l00385" name="l00385"></a><span class="lineno"> 385</span> best_solution_resource_2)</div>
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<div class="line"><a id="l00386" name="l00386"></a><span class="lineno"> 386</span> << std::endl;</div>
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<div class="line"><a id="l00387" name="l00387"></a><span class="lineno"> 387</span> }</div>
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<div class="line"><a id="l00388" name="l00388"></a><span class="lineno"> 388</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a> = path_cost;</div>
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<div class="line"><a id="l00389" name="l00389"></a><span class="lineno"> 389</span> }</div>
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<div class="line"><a id="l00390" name="l00390"></a><span class="lineno"> 390</span> <span class="keywordflow">if</span> (<a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a> < result.objective_value()) {</div>
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<div class="line"><a id="l00391" name="l00391"></a><span class="lineno"> 391</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a> = result.objective_value();</div>
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<div class="line"><a id="l00392" name="l00392"></a><span class="lineno"> 392</span> }</div>
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<div class="line"><a id="l00393" name="l00393"></a><span class="lineno"> 393</span> </div>
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<div class="line"><a id="l00394" name="l00394"></a><span class="lineno"> 394</span> <span class="keywordflow">if</span> (lagrangian_output) {</div>
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<div class="line"><a id="l00395" name="l00395"></a><span class="lineno"> 395</span> std::cout << absl::StrFormat(<span class="stringliteral">"%4d %6.3f %6.3f %9.3f"</span>, iterations,</div>
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<div class="line"><a id="l00396" name="l00396"></a><span class="lineno"> 396</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a>, <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a>, step_size_t)</div>
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<div class="line"><a id="l00397" name="l00397"></a><span class="lineno"> 397</span> << <span class="stringliteral">" "</span> << <a class="code hl_function" href="namespacegtl.html#a252ef610941828aa417152c3230ca670">gtl::LogContainer</a>(mu) << <span class="stringliteral">" "</span></div>
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<div class="line"><a id="l00398" name="l00398"></a><span class="lineno"> 398</span> << <a class="code hl_function" href="namespacegtl.html#a252ef610941828aa417152c3230ca670">gtl::LogContainer</a>(grad_mu_vals) << std::endl;</div>
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<div class="line"><a id="l00399" name="l00399"></a><span class="lineno"> 399</span> }</div>
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<div class="line"><a id="l00400" name="l00400"></a><span class="lineno"> 400</span> </div>
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<div class="line"><a id="l00401" name="l00401"></a><span class="lineno"> 401</span> <span class="comment">// Termination criteria</span></div>
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<div class="line"><a id="l00402" name="l00402"></a><span class="lineno"> 402</span> <span class="keywordtype">double</span> norm = 0;</div>
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<div class="line"><a id="l00403" name="l00403"></a><span class="lineno"> 403</span> <span class="keywordflow">for</span> (<span class="keywordtype">double</span> <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a> : grad_mu_vals) {</div>
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<div class="line"><a id="l00404" name="l00404"></a><span class="lineno"> 404</span> norm += (<a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a> * <a class="code hl_variable" href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a>);</div>
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<div class="line"><a id="l00405" name="l00405"></a><span class="lineno"> 405</span> }</div>
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<div class="line"><a id="l00406" name="l00406"></a><span class="lineno"> 406</span> norm = sqrt(norm);</div>
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<div class="line"><a id="l00407" name="l00407"></a><span class="lineno"> 407</span> <span class="keywordflow">if</span> (iterations == max_iterations || <a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a> == <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a> ||</div>
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<div class="line"><a id="l00408" name="l00408"></a><span class="lineno"> 408</span> step_size_t * norm < <a class="code hl_variable" href="lagrangian__relaxation_8cc.html#a6ffb9a02546adae760787b2366ae6e41">kZeroTol</a>) {</div>
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<div class="line"><a id="l00409" name="l00409"></a><span class="lineno"> 409</span> termination = <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00410" name="l00410"></a><span class="lineno"> 410</span> }</div>
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<div class="line"><a id="l00411" name="l00411"></a><span class="lineno"> 411</span> iterations++;</div>
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<div class="line"><a id="l00412" name="l00412"></a><span class="lineno"> 412</span> }</div>
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<div class="line"><a id="l00413" name="l00413"></a><span class="lineno"> 413</span> </div>
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<div class="line"><a id="l00414" name="l00414"></a><span class="lineno"> 414</span> std::cout << <span class="stringliteral">"Lagrangian relaxation with 2 side constraints"</span> << std::endl;</div>
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<div class="line"><a id="l00415" name="l00415"></a><span class="lineno"> 415</span> std::cout << <span class="stringliteral">"Constraint for resource 1 dualized: "</span></div>
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<div class="line"><a id="l00416" name="l00416"></a><span class="lineno"> 416</span> << (dualized_resource_1 ? <span class="stringliteral">"true"</span> : <span class="stringliteral">"false"</span>) << std::endl;</div>
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<div class="line"><a id="l00417" name="l00417"></a><span class="lineno"> 417</span> std::cout << <span class="stringliteral">"Constraint for resource 2 dualized: "</span></div>
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<div class="line"><a id="l00418" name="l00418"></a><span class="lineno"> 418</span> << (dualized_resource_2 ? <span class="stringliteral">"true"</span> : <span class="stringliteral">"false"</span>) << std::endl;</div>
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<div class="line"><a id="l00419" name="l00419"></a><span class="lineno"> 419</span> std::cout << absl::StrFormat(<span class="stringliteral">"Lower bound: %6.3f"</span>, <a class="code hl_variable" href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a>) << std::endl;</div>
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<div class="line"><a id="l00420" name="l00420"></a><span class="lineno"> 420</span> std::cout << absl::StrFormat(<span class="stringliteral">"Upper bound: %6.3f (Integer solution)"</span>,</div>
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<div class="line"><a id="l00421" name="l00421"></a><span class="lineno"> 421</span> <a class="code hl_variable" href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a>)</div>
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<div class="line"><a id="l00422" name="l00422"></a><span class="lineno"> 422</span> << std::endl;</div>
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<div class="line"><a id="l00423" name="l00423"></a><span class="lineno"> 423</span> std::cout << <span class="stringliteral">"========================================"</span> << std::endl;</div>
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<div class="line"><a id="l00424" name="l00424"></a><span class="lineno"> 424</span>}</div>
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<div class="line"><a id="l00425" name="l00425"></a><span class="lineno"> 425</span> </div>
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<div class="line"><a id="l00426" name="l00426"></a><span class="lineno"> 426</span><span class="keywordtype">void</span> RelaxModel(FlowModel& flow_model) {</div>
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<div class="line"><a id="l00427" name="l00427"></a><span class="lineno"> 427</span> <span class="keywordflow">for</span> (Variable& <a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a> : flow_model.flow_vars) {</div>
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<div class="line"><a id="l00428" name="l00428"></a><span class="lineno"> 428</span> flow_model.model->set_continuous(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>);</div>
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<div class="line"><a id="l00429" name="l00429"></a><span class="lineno"> 429</span> flow_model.model->set_lower_bound(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>, 0.0);</div>
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<div class="line"><a id="l00430" name="l00430"></a><span class="lineno"> 430</span> flow_model.model->set_upper_bound(<a class="code hl_variable" href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a>, 1.0);</div>
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<div class="line"><a id="l00431" name="l00431"></a><span class="lineno"> 431</span> }</div>
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<div class="line"><a id="l00432" name="l00432"></a><span class="lineno"> 432</span>}</div>
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<div class="line"><a id="l00433" name="l00433"></a><span class="lineno"> 433</span> </div>
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<div class="line"><a id="l00434" name="l00434"></a><span class="lineno"> 434</span><span class="keywordtype">void</span> SolveFullModel(<span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a>& graph, <span class="keywordtype">double</span> max_resource_1,</div>
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<div class="line"><a id="l00435" name="l00435"></a><span class="lineno"> 435</span> <span class="keywordtype">double</span> max_resource_2) {</div>
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<div class="line"><a id="l00436" name="l00436"></a><span class="lineno"> 436</span> FlowModel flow_model = SolveMip(graph, max_resource_1, max_resource_2);</div>
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<div class="line"><a id="l00437" name="l00437"></a><span class="lineno"> 437</span> RelaxModel(flow_model);</div>
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<div class="line"><a id="l00438" name="l00438"></a><span class="lineno"> 438</span> SolveLinearRelaxation(flow_model, graph, max_resource_1, max_resource_2);</div>
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<div class="line"><a id="l00439" name="l00439"></a><span class="lineno"> 439</span>}</div>
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<div class="line"><a id="l00440" name="l00440"></a><span class="lineno"> 440</span> </div>
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<div class="line"><a id="l00441" name="l00441"></a><span class="lineno"> 441</span>} <span class="comment">// namespace</span></div>
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<div class="line"><a id="l00442" name="l00442"></a><span class="lineno"> 442</span> </div>
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<div class="line"><a id="l00443" name="l00443"></a><span class="lineno"><a class="line" href="lagrangian__relaxation_8cc.html#a3c04138a5bfe5d72780bb7e82a18e627"> 443</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="lagrangian__relaxation_8cc.html#a3c04138a5bfe5d72780bb7e82a18e627">main</a>(<span class="keywordtype">int</span> argc, <span class="keywordtype">char</span>** argv) {</div>
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<div class="line"><a id="l00444" name="l00444"></a><span class="lineno"> 444</span> <a class="code hl_function" href="namespacegoogle.html#a1749056ff206ebc4f581e6bc0bae841d">google::InitGoogleLogging</a>(argv[0]);</div>
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<div class="line"><a id="l00445" name="l00445"></a><span class="lineno"> 445</span> absl::ParseCommandLine(argc, argv);</div>
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<div class="line"><a id="l00446" name="l00446"></a><span class="lineno"> 446</span> </div>
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<div class="line"><a id="l00447" name="l00447"></a><span class="lineno"> 447</span> <span class="comment">// Problem data</span></div>
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<div class="line"><a id="l00448" name="l00448"></a><span class="lineno"> 448</span> <span class="keyword">const</span> <a class="code hl_typedef" href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">Graph</a> graph = CreateSampleNetwork();</div>
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<div class="line"><a id="l00449" name="l00449"></a><span class="lineno"> 449</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_1 = 10;</div>
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<div class="line"><a id="l00450" name="l00450"></a><span class="lineno"> 450</span> <span class="keyword">const</span> <span class="keywordtype">double</span> max_resource_2 = 4;</div>
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<div class="line"><a id="l00451" name="l00451"></a><span class="lineno"> 451</span> </div>
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<div class="line"><a id="l00452" name="l00452"></a><span class="lineno"> 452</span> SolveFullModel(graph, max_resource_1, max_resource_2);</div>
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<div class="line"><a id="l00453" name="l00453"></a><span class="lineno"> 453</span> </div>
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<div class="line"><a id="l00454" name="l00454"></a><span class="lineno"> 454</span> SolveLagrangianRelaxation(graph, max_resource_1, max_resource_2);</div>
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<div class="line"><a id="l00455" name="l00455"></a><span class="lineno"> 455</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00456" name="l00456"></a><span class="lineno"> 456</span>}</div>
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<div class="ttc" id="aalldiff__cst_8cc_html_ad10edae0a852d72fb76afb1c77735045"><div class="ttname"><a href="alldiff__cst_8cc.html#ad10edae0a852d72fb76afb1c77735045">min</a></div><div class="ttdeci">int64_t min</div><div class="ttdef"><b>Definition:</b> <a href="alldiff__cst_8cc_source.html#l00139">alldiff_cst.cc:139</a></div></div>
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<div class="ttc" id="abase_2logging_8h_html"><div class="ttname"><a href="base_2logging_8h.html">logging.h</a></div></div>
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<div class="ttc" id="abase_2logging_8h_html_a3e1cfef60e774a81f30eaddf26a3a274"><div class="ttname"><a href="base_2logging_8h.html#a3e1cfef60e774a81f30eaddf26a3a274">CHECK</a></div><div class="ttdeci">#define CHECK(condition)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00495">base/logging.h:495</a></div></div>
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<div class="ttc" id="abase_2logging_8h_html_a4bd2e815ca2f702a4b6aa744b1ff3b82"><div class="ttname"><a href="base_2logging_8h.html#a4bd2e815ca2f702a4b6aa744b1ff3b82">CHECK_LT</a></div><div class="ttdeci">#define CHECK_LT(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00705">base/logging.h:705</a></div></div>
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<div class="ttc" id="abase_2logging_8h_html_a7c0ce053b28d53aa4eaf3eb7fb71663b"><div class="ttname"><a href="base_2logging_8h.html#a7c0ce053b28d53aa4eaf3eb7fb71663b">CHECK_EQ</a></div><div class="ttdeci">#define CHECK_EQ(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00702">base/logging.h:702</a></div></div>
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<div class="ttc" id="abase_2logging_8h_html_a7e03ec13560fa94a8fea569960d7efc6"><div class="ttname"><a href="base_2logging_8h.html#a7e03ec13560fa94a8fea569960d7efc6">CHECK_GT</a></div><div class="ttdeci">#define CHECK_GT(val1, val2)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8h_source.html#l00707">base/logging.h:707</a></div></div>
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<div class="ttc" id="acontainer__logging_8h_html"><div class="ttname"><a href="container__logging_8h.html">container_logging.h</a></div></div>
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<div class="ttc" id="ademon__profiler_8cc_html_ac072af30c4ffbc834bb4c681f6ecb514"><div class="ttname"><a href="demon__profiler_8cc.html#ac072af30c4ffbc834bb4c681f6ecb514">value</a></div><div class="ttdeci">int64_t value</div><div class="ttdef"><b>Definition:</b> <a href="demon__profiler_8cc_source.html#l00044">demon_profiler.cc:44</a></div></div>
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<div class="ttc" id="aexpr__array_8cc_html_a472a99923cbe11ae7b5a5d157d9ad465"><div class="ttname"><a href="expr__array_8cc.html#a472a99923cbe11ae7b5a5d157d9ad465">var</a></div><div class="ttdeci">IntVar * var</div><div class="ttdef"><b>Definition:</b> <a href="expr__array_8cc_source.html#l01874">expr_array.cc:1874</a></div></div>
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<div class="ttc" id="agscip__solver_8cc_html_a1ba5ca0f61f2fa13bd23bf0f89004f35"><div class="ttname"><a href="gscip__solver_8cc.html#a1ba5ca0f61f2fa13bd23bf0f89004f35">upper_bound</a></div><div class="ttdeci">double upper_bound</div><div class="ttdef"><b>Definition:</b> <a href="gscip__solver_8cc_source.html#l00137">gscip_solver.cc:137</a></div></div>
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<div class="ttc" id="agscip__solver_8cc_html_a1e2f9a2352c1d9a6cada9544898fceec"><div class="ttname"><a href="gscip__solver_8cc.html#a1e2f9a2352c1d9a6cada9544898fceec">lower_bound</a></div><div class="ttdeci">double lower_bound</div><div class="ttdef"><b>Definition:</b> <a href="gscip__solver_8cc_source.html#l00136">gscip_solver.cc:136</a></div></div>
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<div class="ttc" id="agurobi__interface_8cc_html_a0728f23c9a47655d38e0bf1a2f200bcf"><div class="ttname"><a href="gurobi__interface_8cc.html#a0728f23c9a47655d38e0bf1a2f200bcf">model</a></div><div class="ttdeci">GRBmodel * model</div><div class="ttdef"><b>Definition:</b> <a href="gurobi__interface_8cc_source.html#l00274">gurobi_interface.cc:274</a></div></div>
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<div class="ttc" id="alagrangian__relaxation_8cc_html_a3c04138a5bfe5d72780bb7e82a18e627"><div class="ttname"><a href="lagrangian__relaxation_8cc.html#a3c04138a5bfe5d72780bb7e82a18e627">main</a></div><div class="ttdeci">int main(int argc, char **argv)</div><div class="ttdef"><b>Definition:</b> <a href="lagrangian__relaxation_8cc_source.html#l00443">lagrangian_relaxation.cc:443</a></div></div>
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<div class="ttc" id="alagrangian__relaxation_8cc_html_a3dd301d5a8f137443e3619bd9882b23a"><div class="ttname"><a href="lagrangian__relaxation_8cc.html#a3dd301d5a8f137443e3619bd9882b23a">ABSL_FLAG</a></div><div class="ttdeci">ABSL_FLAG(double, step_size, 0.95, "Stepsize for gradient ascent, determined as step_size^t.")</div></div>
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<div class="ttc" id="alagrangian__relaxation_8cc_html_a6ffb9a02546adae760787b2366ae6e41"><div class="ttname"><a href="lagrangian__relaxation_8cc.html#a6ffb9a02546adae760787b2366ae6e41">kZeroTol</a></div><div class="ttdeci">constexpr double kZeroTol</div><div class="ttdef"><b>Definition:</b> <a href="lagrangian__relaxation_8cc_source.html#l00113">lagrangian_relaxation.cc:113</a></div></div>
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<div class="ttc" id="amath__opt_8h_html"><div class="ttname"><a href="math__opt_8h.html">math_opt.h</a></div></div>
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<div class="ttc" id="amathutil_8h_html"><div class="ttname"><a href="mathutil_8h.html">mathutil.h</a></div></div>
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<div class="ttc" id="anamespacegoogle_html_a1749056ff206ebc4f581e6bc0bae841d"><div class="ttname"><a href="namespacegoogle.html#a1749056ff206ebc4f581e6bc0bae841d">google::InitGoogleLogging</a></div><div class="ttdeci">void InitGoogleLogging(const char *argv0)</div><div class="ttdef"><b>Definition:</b> <a href="base_2logging_8cc_source.html#l01873">base/logging.cc:1873</a></div></div>
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<div class="ttc" id="anamespacegtl_html_a252ef610941828aa417152c3230ca670"><div class="ttname"><a href="namespacegtl.html#a252ef610941828aa417152c3230ca670">gtl::LogContainer</a></div><div class="ttdeci">auto LogContainer(const ContainerT &container, const PolicyT &policy) -> decltype(gtl::LogRange(container.begin(), container.end(), policy))</div><div class="ttdef"><b>Definition:</b> <a href="container__logging_8h_source.html#l00275">container_logging.h:275</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_1_1math__opt_html_a252c66967569c5ab1db4b4d356707fb1"><div class="ttname"><a href="namespaceoperations__research_1_1math__opt.html#a252c66967569c5ab1db4b4d356707fb1">operations_research::math_opt::VariableMap</a></div><div class="ttdeci">IdMap< Variable, V > VariableMap</div><div class="ttdef"><b>Definition:</b> <a href="variable__and__expressions_8h_source.html#l00151">variable_and_expressions.h:151</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_1_1math__opt_html_a5767ee23f380e72488d3c7ebf2d742b1"><div class="ttname"><a href="namespaceoperations__research_1_1math__opt.html#a5767ee23f380e72488d3c7ebf2d742b1">operations_research::math_opt::Solve</a></div><div class="ttdeci">absl::StatusOr< SolveResult > Solve(const Model &model, const SolverType solver_type, const SolveArguments &solve_args, const SolverInitArguments &init_args)</div><div class="ttdef"><b>Definition:</b> <a href="solve_8cc_source.html#l00155">solve.cc:155</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_1_1math__opt_html_a8b06041d7c1fb05f379714f4312306ec"><div class="ttname"><a href="namespaceoperations__research_1_1math__opt.html#a8b06041d7c1fb05f379714f4312306ec">operations_research::math_opt::SolverType</a></div><div class="ttdeci">SolverType</div><div class="ttdef"><b>Definition:</b> <a href="parameters_8h_source.html#l00035">parameters.h:35</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_1_1math__opt_html_ad02e69a0531469b463df907c7b2ad194"><div class="ttname"><a href="namespaceoperations__research_1_1math__opt.html#ad02e69a0531469b463df907c7b2ad194">operations_research::math_opt::TerminationReason</a></div><div class="ttdeci">TerminationReason</div><div class="ttdef"><b>Definition:</b> <a href="solve__result_8h_source.html#l00148">solve_result.h:148</a></div></div>
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<div class="ttc" id="anamespaceoperations__research_html_a7deeae530369e107f9d91c1a189f451f"><div class="ttname"><a href="namespaceoperations__research.html#a7deeae530369e107f9d91c1a189f451f">operations_research::Arc</a></div><div class="ttdeci">std::pair< int64_t, int64_t > Arc</div><div class="ttdef"><b>Definition:</b> <a href="search_8cc_source.html#l03434">search.cc:3434</a></div></div>
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<div class="ttc" id="anamespaceutil_html_a2f76166dbe0c4055a1f256235ad00478"><div class="ttname"><a href="namespaceutil.html#a2f76166dbe0c4055a1f256235ad00478">util::Graph</a></div><div class="ttdeci">ListGraph Graph</div><div class="ttdef"><b>Definition:</b> <a href="graph_8h_source.html#l02362">graph.h:2362</a></div></div>
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<div class="ttc" id="arouting__flow_8cc_html_a75d7b5e4cab1e156cc7a2c5eba1e16f1"><div class="ttname"><a href="routing__flow_8cc.html#a75d7b5e4cab1e156cc7a2c5eba1e16f1">cost</a></div><div class="ttdeci">int64_t cost</div><div class="ttdef"><b>Definition:</b> <a href="routing__flow_8cc_source.html#l00152">routing_flow.cc:152</a></div></div>
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