A computational study for bilevel quadratic programs using semidefinite relaxations

•We consider bilevel quadratic problems with binary variables in the leader and convex quadratic follower problems.•We derive equivalent Mixed Integer Linear Programming formulations. Thus, we compute optimal solutions and upper bounds.•We transform the bilevel problems into binary quadratic program...

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Vydáno v:European journal of operational research Ročník 254; číslo 1; s. 9 - 18
Hlavní autoři: Adasme, Pablo, Lisser, Abdel
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.10.2016
Elsevier Sequoia S.A
Elsevier
Témata:
(I) Conic programming and interior point methods
Bilevel programming
Computational efficiency
Computer Science
Convex analysis
Decision analysis
Equivalence
Followers
Integer programming
Linear programming
Mathematical models
Mathematical problems
Mixed integer
Mixed integer linear programming
Operations Research
Optimization
Quadratic programming
Semidefinite programming
Studies
Bilevel programming
Semidefinite programming
(I) Conic programming and interior point methods
Mixed integer linear programming
ISSN:0377-2217, 1872-6860
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Popis
Shrnutí:•We consider bilevel quadratic problems with binary variables in the leader and convex quadratic follower problems.•We derive equivalent Mixed Integer Linear Programming formulations. Thus, we compute optimal solutions and upper bounds.•We transform the bilevel problems into binary quadratic programs and derive semidefinite relaxations.•The particular case where the follower problem is formulated as a linear program is also considered.•The SDP bounds are significantly tight. Finally, they are obtained at low computational cost. In this paper, we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem. For this purpose, we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent Karush Kuhn Tucker (KKT) conditions. Then, we use the single level formulations to obtain mixed integer linear programming (MILP) models and semidefinite programming (SDP) relaxations. Thus, we compute optimal solutions and upper bounds using linear programming (LP) and SDP relaxations. Our numerical results indicate that the SDP relaxations are considerably tighter than the LP ones. Consequently, the SDP relaxations allow finding tight feasible solutions for the problem. Especially, when the number of variables in the leader problem is larger than in the follower problem. Moreover, they are solved at a significantly lower computational cost for large scale instances.
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ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2016.01.020