<ahref="initial__basis_8h.html">Go to the documentation of this file.</a><divclass="fragment"><divclass="line"><aid="l00001"name="l00001"></a><spanclass="lineno"> 1</span><spanclass="comment">// Copyright 2010-2021 Google LLC</span></div>
<divclass="line"><aid="l00002"name="l00002"></a><spanclass="lineno"> 2</span><spanclass="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div>
<divclass="line"><aid="l00003"name="l00003"></a><spanclass="lineno"> 3</span><spanclass="comment">// you may not use this file except in compliance with the License.</span></div>
<divclass="line"><aid="l00004"name="l00004"></a><spanclass="lineno"> 4</span><spanclass="comment">// You may obtain a copy of the License at</span></div>
<divclass="line"><aid="l00008"name="l00008"></a><spanclass="lineno"> 8</span><spanclass="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
<divclass="line"><aid="l00009"name="l00009"></a><spanclass="lineno"> 9</span><spanclass="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div>
<divclass="line"><aid="l00010"name="l00010"></a><spanclass="lineno"> 10</span><spanclass="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
<divclass="line"><aid="l00011"name="l00011"></a><spanclass="lineno"> 11</span><spanclass="comment">// See the License for the specific language governing permissions and</span></div>
<divclass="line"><aid="l00012"name="l00012"></a><spanclass="lineno"> 12</span><spanclass="comment">// limitations under the License.</span></div>
<divclass="line"><aid="l00024"name="l00024"></a><spanclass="lineno"> 24</span><spanclass="comment">// This class implements two initial basis algorithms. The idea is to replace as</span></div>
<divclass="line"><aid="l00025"name="l00025"></a><spanclass="lineno"> 25</span><spanclass="comment">// much as possible the columns of B that correspond to fixed slack variables</span></div>
<divclass="line"><aid="l00026"name="l00026"></a><spanclass="lineno"> 26</span><spanclass="comment">// with some column of A in order to have more freedom in the values the basic</span></div>
<divclass="line"><aid="l00027"name="l00027"></a><spanclass="lineno"> 27</span><spanclass="comment">// variables can take.</span></div>
<divclass="line"><aid="l00029"name="l00029"></a><spanclass="lineno"> 29</span><spanclass="comment">// The first algorithm is Bixby's initial basis algorithm, described in the</span></div>
<divclass="line"><aid="l00030"name="l00030"></a><spanclass="lineno"> 30</span><spanclass="comment">// paper below. It considers the columns of A in a particular order (the ones</span></div>
<divclass="line"><aid="l00031"name="l00031"></a><spanclass="lineno"> 31</span><spanclass="comment">// with more freedom first) and adds the current column to the basis if it keeps</span></div>
<divclass="line"><aid="l00032"name="l00032"></a><spanclass="lineno"> 32</span><spanclass="comment">// B almost triangular and with coefficients close to 1.0 on the diagonal for</span></div>
<divclass="line"><aid="l00033"name="l00033"></a><spanclass="lineno"> 33</span><spanclass="comment">// good numerical stability.</span></div>
<divclass="line"><aid="l00035"name="l00035"></a><spanclass="lineno"> 35</span><spanclass="comment">// Robert E. Bixby, "Implementing the Simplex Method: The Initial Basis"</span></div>
<divclass="line"><aid="l00039"name="l00039"></a><spanclass="lineno"> 39</span><spanclass="comment">// The second algorithm is is similar to the "advanced initial basis" that GLPK</span></div>
<divclass="line"><aid="l00040"name="l00040"></a><spanclass="lineno"> 40</span><spanclass="comment">// uses by default. It adds columns one by one to the basis B while keeping it</span></div>
<divclass="line"><aid="l00041"name="l00041"></a><spanclass="lineno"> 41</span><spanclass="comment">// triangular (not almost triangular as in Bixby's algorithm). The next</span></div>
<divclass="line"><aid="l00042"name="l00042"></a><spanclass="lineno"> 42</span><spanclass="comment">// column to add is chosen amongst the set of possible candidates using a</span></div>
<divclass="line"><aid="l00043"name="l00043"></a><spanclass="lineno"> 43</span><spanclass="comment">// heuristic similar to the one used by Bixby.</span></div>
<divclass="line"><aid="l00046"name="l00046"></a><spanclass="lineno"> 46</span><spanclass="comment">// Takes references to the linear program data we need.</span></div>
<divclass="line"><aid="l00052"name="l00052"></a><spanclass="lineno"> 52</span><spanclass="comment">// Completes the entries of the given basis that are equal to kInvalidCol with</span></div>
<divclass="line"><aid="l00053"name="l00053"></a><spanclass="lineno"> 53</span><spanclass="comment">// one of the first num_cols columns of A using Bixby's algorithm.</span></div>
<divclass="line"><aid="l00055"name="l00055"></a><spanclass="lineno"> 55</span><spanclass="comment">// Important: For this function, the matrix must be scaled such that the</span></div>
<divclass="line"><aid="l00056"name="l00056"></a><spanclass="lineno"> 56</span><spanclass="comment">// maximum absolute value in each column is 1.0.</span></div>
<divclass="line"><aid="l00059"name="l00059"></a><spanclass="lineno"> 59</span><spanclass="comment">// Similar to CompleteBixbyBasis() but completes the basis into a triangular</span></div>
<divclass="line"><aid="l00060"name="l00060"></a><spanclass="lineno"> 60</span><spanclass="comment">// one. This function usually produces better initial bases. The dual version</span></div>
<divclass="line"><aid="l00061"name="l00061"></a><spanclass="lineno"> 61</span><spanclass="comment">// just restricts the possible entering columns to the ones with a cost of 0.0</span></div>
<divclass="line"><aid="l00062"name="l00062"></a><spanclass="lineno"> 62</span><spanclass="comment">// in order to always start with the all-zeros vector of dual values.</span></div>
<divclass="line"><aid="l00064"name="l00064"></a><spanclass="lineno"> 64</span><spanclass="comment">// Returns false if an error occurred during the algorithm (numerically</span></div>
<divclass="line"><aid="l00069"name="l00069"></a><spanclass="lineno"> 69</span><spanclass="comment">// Use Maros's LTSF crash from the book "Computational Techniques of the</span></div>
<divclass="line"><aid="l00070"name="l00070"></a><spanclass="lineno"> 70</span><spanclass="comment">// Simplex Method". Unlike the other crashes this does not use the initial</span></div>
<divclass="line"><aid="l00071"name="l00071"></a><spanclass="lineno"> 71</span><spanclass="comment">// content of the basis parameter.</span></div>
<divclass="line"><aid="l00075"name="l00075"></a><spanclass="lineno"> 75</span><spanclass="comment">// Visible for testing. Computes a list of candidate column indices out of the</span></div>
<divclass="line"><aid="l00076"name="l00076"></a><spanclass="lineno"> 76</span><spanclass="comment">// fist num_candidate_columns of A and sorts them using the</span></div>
<divclass="line"><aid="l00077"name="l00077"></a><spanclass="lineno"> 77</span><spanclass="comment">// bixby_column_comparator_. This also fills max_scaled_abs_cost_.</span></div>
<divclass="line"><aid="l00081"name="l00081"></a><spanclass="lineno"> 81</span><spanclass="comment">// Internal implementation of the Primal/Dual CompleteTriangularBasis().</span></div>
<divclass="line"><aid="l00088"name="l00088"></a><spanclass="lineno"> 88</span><spanclass="comment">// Returns an integer representing the order (the lower the better)</span></div>
<divclass="line"><aid="l00089"name="l00089"></a><spanclass="lineno"> 89</span><spanclass="comment">// between column categories (known as C2, C3 or C4 in the paper).</span></div>
<divclass="line"><aid="l00090"name="l00090"></a><spanclass="lineno"> 90</span><spanclass="comment">// Also returns a greater index for fixed columns.</span></div>
<divclass="line"><aid="l00093"name="l00093"></a><spanclass="lineno"> 93</span><spanclass="comment">// Row and column priorities for Maros crash.</span></div>
<divclass="line"><aid="l00097"name="l00097"></a><spanclass="lineno"> 97</span><spanclass="comment">// Returns the penalty (the lower the better) of a column. This is 'q_j' for a</span></div>
<divclass="line"><aid="l00098"name="l00098"></a><spanclass="lineno"> 98</span><spanclass="comment">// column 'j' in the paper.</span></div>
<divclass="line"><aid="l00101"name="l00101"></a><spanclass="lineno"> 101</span><spanclass="comment">// Maximum scaled absolute value of the objective for the columns which are</span></div>
<divclass="line"><aid="l00102"name="l00102"></a><spanclass="lineno"> 102</span><spanclass="comment">// entering candidates. This is used by GetColumnPenalty().</span></div>
<divclass="line"><aid="l00105"name="l00105"></a><spanclass="lineno"> 105</span><spanclass="comment">// Comparator used to sort column indices according to their penalty.</span></div>
<divclass="line"><aid="l00106"name="l00106"></a><spanclass="lineno"> 106</span><spanclass="comment">// Lower is better.</span></div>
<divclass="line"><aid="l00114"name="l00114"></a><spanclass="lineno"> 114</span><spanclass="comment">// Comparator used by CompleteTriangularBasis(). Note that this one is meant</span></div>
<divclass="line"><aid="l00115"name="l00115"></a><spanclass="lineno"> 115</span><spanclass="comment">// to be used by a priority queue, so higher is better.</span></div>
<divclass="ttc"id="anamespaceoperations__research_html"><divclass="ttname"><ahref="namespaceoperations__research.html">operations_research</a></div><divclass="ttdoc">Collection of objects used to extend the Constraint Solver library.</div><divclass="ttdef"><b>Definition:</b><ahref="dense__doubly__linked__list_8h_source.html#l00021">dense_doubly_linked_list.h:21</a></div></div>
