scholarly journals A Branch-and-cut Algorithm for Integer Bilevel Linear Programs

2009 ◽  
pp. 65-78 ◽  
Author(s):  
S. T. DeNegre ◽  
T. K. Ralphs
Keyword(s):  
Branch And Cut ◽  
Linear Programs ◽  
Bilevel Linear Programs

scholarly journals A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

2017 ◽  
Vol 65 (6) ◽  
pp. 1615-1637 ◽  
Author(s):  
Matteo Fischetti ◽  
Ivana Ljubić ◽  
Michele Monaci ◽  
Markus Sinnl
Keyword(s):  
General Purpose ◽  
Mixed Integer ◽  
Linear Programs ◽  
Bilevel Linear Programs

scholarly journals Branch-and-cut for linear programs with overlapping SOS1 constraints

2017 ◽  
Vol 10 (1) ◽  
pp. 33-68 ◽  
Author(s):  
Tobias Fischer ◽  
Marc E. Pfetsch
Keyword(s):  
Branch And Cut ◽  
Linear Programs

Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut

1991 ◽  
Vol 3 (2) ◽  
pp. 121-134 ◽  
Author(s):  
Karla L. Hoffman ◽  
Manfred Padberg
Keyword(s):  
Branch And Cut ◽  
Linear Programs

scholarly journals Consistency Cuts for Dantzig-Wolfe Reformulations

2022 ◽  
Author(s):  
Jens Vinther Clausen ◽  
Richard Lusby ◽  
Stefan Ropke
Keyword(s):  
Valid Inequalities ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Linear Programming Relaxation ◽  
Linear Programs ◽  
Integer Programs ◽  
New Family ◽  
Integer Linear Programs ◽  
Integer Solutions ◽  
Online Library

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

scholarly journals Scenario Grouping and Decomposition Algorithms for Chance-Constrained Programs

2020 ◽  
Author(s):  
Yan Deng ◽  
Huiwen Jia ◽  
Shabbir Ahmed ◽  
Jon Lee ◽  
Siqian Shen
Keyword(s):  
Operations Research ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Decomposition Algorithms ◽  
Linear Programs ◽  
Solution Quality ◽  
Bilevel Program ◽  
Research Problems ◽  
Optimal Grouping

A lower bound for a finite-scenario-based chance-constrained program is the quantile value corresponding to the sorted optimal objective values of scenario subproblems. This quantile bound can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. The quality of the bound depends on how the scenarios are grouped. In this paper, we formulate a mixed-integer bilevel program that optimally groups scenarios to tighten the quantile bounds. For general chance-constrained programs, we propose a branch-and-cut algorithm to optimize the bilevel program, and for chance-constrained linear programs, a mixed-integer linear-programming reformulation is derived. We also propose several heuristics for grouping similar or dissimilar scenarios. Our computational results demonstrate that optimal grouping bounds are much tighter than heuristic bounds, resulting in smaller root-node gaps and better performance of scenario decomposition for solving chance-constrained 0-1 programs. Also, the optimal grouping bounds can be greatly strengthened using larger group size. Summary of Contribution: Chance-constrained programs are in general NP-hard but widely used in practice for lowering the risk of undesirable outcomes during decision making under uncertainty. Assuming finite scenarios of uncertain parameter, chance-constrained programs can be reformulated as mixed-integer linear programs with binary variables representing whether or not the constraints are satisfied in corresponding scenarios. A useful quantile bound for solving chance-constrained programs can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. In this paper, we develop algorithms for optimally and heuristically grouping scenarios to tighten the quantile bounds. We aim to improve both the computation and solution quality of a variety of chance-constrained programs formulated for different Operations Research problems.

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