scholarly journals A computational status update for exact rational mixed integer programming

2022 ◽  
Author(s):  
Leon Eifler ◽  
Ambros Gleixner
Keyword(s):  
Rational Numbers ◽  
Iterative Refinement ◽  
Mixed Integer ◽  
Roundoff Error ◽  
Integer Programs ◽  
Mixed Integer Programs ◽  
Primal Heuristics ◽  
Improved Performance ◽  
Computational Behavior ◽  
Exact Repair

AbstractThe last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.

scholarly journals Learning Primal Heuristics for Mixed Integer Programs

2021 ◽  
Author(s):  
Yunzhuang Shen ◽  
Yuan Sun ◽  
Andrew Eberhard ◽  
Xiaodong Li
Keyword(s):  
Mixed Integer ◽  
Integer Programs ◽  
Mixed Integer Programs ◽  
Primal Heuristics

Support Set Invariancy for Interval Bimatrix Games

2019 ◽  
Vol 27 (02) ◽  
pp. 225-237
Author(s):  
Milan Hladík
Keyword(s):  
Mixed Integer ◽  
Necessary Condition ◽  
Linear Programs ◽  
Bimatrix Games ◽  
Bimatrix Game ◽  
Integer Programs ◽  
Mixed Integer Programs ◽  
Support Set ◽  
First Case ◽  
Pure Strategies

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.

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