Universität zu Köln

Buchheim, Christoph ORCID: 0000-0001-9974-404X, Liers, Frauke and Oswald, Marcus (2010). Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization. Mathematical Programming B, 124 (1-2). pp. 513-535. Springer.

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Abstract

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP).Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming.Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvatal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Buchheim, Christoph UNSPECIFIED orcid.org/0000-0001-9974-404X UNSPECIFIED Liers, Frauke UNSPECIFIED UNSPECIFIED UNSPECIFIED Oswald, Marcus UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-549798
Journal or Publication Title:	Mathematical Programming B
Volume:	124
Number:	1-2
Page Range:	pp. 513-535
Date:	2010
Publisher:	Springer
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Institute of Computer Science
Subjects:	Data processing Computer science
Refereed:	No
URI:	http://kups.ub.uni-koeln.de/id/eprint/54979