Exploiting symmetry in copositive programs via semidefinite hierarchies

• Published in: Mathematical programming Vol. 151; no. 2; pp. 659 - 680
• Main Authors: Dobre, Cristian, Vera, Juan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-07-2015; Springer Nature B.V
• Subjects: Algebra; Approximation; Assignment problem; Biometris (WU MAT); Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Hierarchies; Mathematical analysis; Mathematical and Computational Physics; Mathematical and Statistical Methods - Biometris; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Numerical Analysis; Optimization; Orbits; PE&RC; Semidefinite programming; Studies; Symmetry; Theoretical; Traveling salesman problem; Wiskundige en Statistische Methoden - Biometris; 90C20; 90C26
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-015-0879-0

Copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization. An important line of research is to use semidefinite programming to approximate conic programming over the copositive cone. Two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs, and the lack of information about the quality of the semidefinite programming approximations. These drawbacks are an inevitable consequence of the intractability of the generic problems that such approaches attempt to solve. To address such drawbacks, we develop customized solution approaches for highly symmetric copositive programs, which arise naturally in several contexts. For instance, symmetry properties of combinatorial problems are typically inherited when they are addressed via copositive programming. As a result we are able to compute new bounds for crossing number instances in complete bipartite graphs.