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<divclass="title">initial_basis.h</div></div>
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<ahref="initial__basis_8h.html">Go to the documentation of this file.</a><divclass="fragment"><divclass="line"><aname="l00001"></a><spanclass="lineno"> 1</span> <spanclass="comment">// Copyright 2010-2018 Google LLC</span></div>
<divclass="line"><aname="l00002"></a><spanclass="lineno"> 2</span> <spanclass="comment">// Licensed under the Apache License, Version 2.0 (the "License");</span></div>
<divclass="line"><aname="l00003"></a><spanclass="lineno"> 3</span> <spanclass="comment">// you may not use this file except in compliance with the License.</span></div>
<divclass="line"><aname="l00004"></a><spanclass="lineno"> 4</span> <spanclass="comment">// You may obtain a copy of the License at</span></div>
<divclass="line"><aname="l00008"></a><spanclass="lineno"> 8</span> <spanclass="comment">// Unless required by applicable law or agreed to in writing, software</span></div>
<divclass="line"><aname="l00009"></a><spanclass="lineno"> 9</span> <spanclass="comment">// distributed under the License is distributed on an "AS IS" BASIS,</span></div>
<divclass="line"><aname="l00010"></a><spanclass="lineno"> 10</span> <spanclass="comment">// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span></div>
<divclass="line"><aname="l00011"></a><spanclass="lineno"> 11</span> <spanclass="comment">// See the License for the specific language governing permissions and</span></div>
<divclass="line"><aname="l00012"></a><spanclass="lineno"> 12</span> <spanclass="comment">// limitations under the License.</span></div>
<divclass="line"><aname="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"><aname="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"><aname="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"><aname="l00027"></a><spanclass="lineno"> 27</span> <spanclass="comment">// variables can take.</span></div>
<divclass="line"><aname="l00029"></a><spanclass="lineno"> 29</span> <spanclass="comment">// The first algorithm is Bixby's initial basis algorithm, described in the</span></div>
<divclass="line"><aname="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"><aname="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"><aname="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"><aname="l00033"></a><spanclass="lineno"> 33</span> <spanclass="comment">// good numerical stability.</span></div>
<divclass="line"><aname="l00035"></a><spanclass="lineno"> 35</span> <spanclass="comment">// Robert E. Bixby, "Implementing the Simplex Method: The Initial Basis"</span></div>
<divclass="line"><aname="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"><aname="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"><aname="l00041"></a><spanclass="lineno"> 41</span> <spanclass="comment">// triangular (not almost triangular as in Bixby's algorithm). The next</span></div>
<divclass="line"><aname="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"><aname="l00043"></a><spanclass="lineno"> 43</span> <spanclass="comment">// heuristic similar to the one used by Bixby.</span></div>
<divclass="line"><aname="l00046"></a><spanclass="lineno"> 46</span> <spanclass="comment">// Takes references to the linear program data we need.</span></div>
<divclass="line"><aname="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"><aname="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"><aname="l00055"></a><spanclass="lineno"> 55</span> <spanclass="comment">// Important: For this function, the matrix must be scaled such that the</span></div>
<divclass="line"><aname="l00056"></a><spanclass="lineno"> 56</span> <spanclass="comment">// maximum absolute value in each column is 1.0.</span></div>
<divclass="line"><aname="l00059"></a><spanclass="lineno"> 59</span> <spanclass="comment">// Similar to CompleteBixbyBasis() but completes the basis into a triangular</span></div>
<divclass="line"><aname="l00060"></a><spanclass="lineno"> 60</span> <spanclass="comment">// one. This function usually produces better initial bases. The dual version</span></div>
<divclass="line"><aname="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"><aname="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"><aname="l00064"></a><spanclass="lineno"> 64</span> <spanclass="comment">// Returns false if an error occurred during the algorithm (numerically</span></div>
<divclass="line"><aname="l00069"></a><spanclass="lineno"> 69</span> <spanclass="comment">// Use Maros's LTSF crash from the book "Computational Techniques of the</span></div>
<divclass="line"><aname="l00070"></a><spanclass="lineno"> 70</span> <spanclass="comment">// Simplex Method". Unlike the other crashes this does not use the initial</span></div>
<divclass="line"><aname="l00071"></a><spanclass="lineno"> 71</span> <spanclass="comment">// content of the basis parameter.</span></div>
<divclass="line"><aname="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"><aname="l00076"></a><spanclass="lineno"> 76</span> <spanclass="comment">// fist num_candidate_columns of A and sorts them using the</span></div>
<divclass="line"><aname="l00077"></a><spanclass="lineno"> 77</span> <spanclass="comment">// bixby_column_comparator_. This also fills max_scaled_abs_cost_.</span></div>
<divclass="line"><aname="l00081"></a><spanclass="lineno"> 81</span> <spanclass="comment">// Internal implementation of the Primal/Dual CompleteTriangularBasis().</span></div>
<divclass="line"><aname="l00088"></a><spanclass="lineno"> 88</span> <spanclass="comment">// Returns an integer representing the order (the lower the better)</span></div>
<divclass="line"><aname="l00089"></a><spanclass="lineno"> 89</span> <spanclass="comment">// between column categories (known as C2, C3 or C4 in the paper).</span></div>
<divclass="line"><aname="l00090"></a><spanclass="lineno"> 90</span> <spanclass="comment">// Also returns a greater index for fixed columns.</span></div>
<divclass="line"><aname="l00093"></a><spanclass="lineno"> 93</span> <spanclass="comment">// Row and column priorities for Maros crash.</span></div>
<divclass="line"><aname="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"><aname="l00098"></a><spanclass="lineno"> 98</span> <spanclass="comment">// column 'j' in the paper.</span></div>
<divclass="line"><aname="l00101"></a><spanclass="lineno"> 101</span> <spanclass="comment">// Maximum scaled absolute value of the objective for the columns which are</span></div>
<divclass="line"><aname="l00102"></a><spanclass="lineno"> 102</span> <spanclass="comment">// entering candidates. This is used by GetColumnPenalty().</span></div>
<divclass="line"><aname="l00105"></a><spanclass="lineno"> 105</span> <spanclass="comment">// Comparator used to sort column indices according to their penalty.</span></div>
<divclass="line"><aname="l00106"></a><spanclass="lineno"> 106</span> <spanclass="comment">// Lower is better.</span></div>
<divclass="line"><aname="l00114"></a><spanclass="lineno"> 114</span> <spanclass="comment">// Comparator used by CompleteTriangularBasis(). Note that this one is meant</span></div>
<divclass="line"><aname="l00115"></a><spanclass="lineno"> 115</span> <spanclass="comment">// to be used by a priority queue, so higher is better.</span></div>
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