On the accuracy of uniform polyhedral approximations of the copositive cone

Date
2012
Authors
Yıldırım, A.
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Source Title
Optimization Methods and Software
Print ISSN
1055-6788
Electronic ISSN
1029-4937
Publisher
Taylor & Francis
Volume
27
Issue
1
Pages
155 - 173
Language
English
Type
Article
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Abstract

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit. By combining our outer polyhedral approximations with the inner polyhedral approximations due to de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp. 875-892], we obtain a sequence of increasingly sharper lower and upper bounds on the optimal value of a copositive program. Under primal and dual regularity assumptions, we establish that both sequences converge to the optimal value. For standard quadratic optimization problems, we derive tight bounds on the gap between the upper and lower bounds. We provide closed-form expressions of the bounds for the maximum stable set problem. Our computational results shed light on the quality of the bounds on randomly generated instances.

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Keywords
Copositive cone, Completely positive cone, Conic optimization, Standard quadratic optimization, Optimization problems
Citation
Published Version (Please cite this version)