fix: improve the lexico-minimum finder in LemkeAlgorithm. #4295
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The following issues in btLemkeAlgorithm::findLexicographicMinimum() are addressed in this commit.
Logic Issues
and if it contains z_0, the tie must be broken in favor of z_0 and return,
which is not performed in the original.
Efficiency Issues
Description of the Fix
The improved version performs the lexico-minimum test column-by-column
from the left-most column of (-B^{-1}q:B^{-1}).
If the test finds a single row, it returns it.
If the test finds multiple rows and if they contain z_0, it returns z_0.
Otherwise, it moves on to the next column.
Please see 2.2.2 "Pivot Step" of
K. Murty, "Linear Complementarity, Linear and Nonlinear Programming" Heldermann Verlag, Berlin, 1998
for the algorithm details.
Test Results
Some performance tests are done and non-negligible improvement was observed.
Please see https://github.com/ShoYamanishi/AppleNumericalComputing/tree/main/14_lcp for the test details.