Exploiting symmetry in copositive programs via semidefinite hierarchies

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...

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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
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Abstract	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.
AbstractList	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. Issue Title: Special Issue: Polynomial Optimization 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.
Author	Dobre, Cristian Vera, Juan
Author_xml	– sequence: 1 givenname: Cristian surname: Dobre fullname: Dobre, Cristian organization: Biometris, Wageningen University and Research Center – sequence: 2 givenname: Juan surname: Vera fullname: Vera, Juan email: j.c.veralizcano@tilburguniversity.edu organization: Department of Econometrics and OR, Tilburg University
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Optim.20061641076109110.1137/03060151X1131.900402219133 de KlerkEDobreCPasechnikDVNumerical block diag E Klerk de (879_CR37) 2008; 19 879_CR12 C Bachoc (879_CR3) 2008; 21 879_CR5 AJ Quist (879_CR55) 1998; 9 E Klerk de (879_CR33) 2012; 136 879_CR18 E Klerk de (879_CR38) 2014; 233 IM Bomze (879_CR10) 2011; 3 C Riener (879_CR54) 2013; 38 E Klerk de (879_CR35) 2002; 12 RH Tutuncu (879_CR61) 2003; 95 M Dür (879_CR22) 2010 879_CR50 H Dong (879_CR20) 2013; 137 IM Bomze (879_CR8) 2010; 124 P Turán (879_CR60) 1977; 1 Y Kanno (879_CR27) 2001; 2 G Pólya (879_CR51) 1928; 73 S Burer (879_CR15) 2009; 120 J Chen (879_CR17) 2012; 4 DJ Kleitman (879_CR28) 1970; 9 J Povh (879_CR53) 2007; 18 S Bundfuss (879_CR13) 2009; 428 H Dong (879_CR19) 2013; 23 A Schrijver (879_CR57) 2005; 51 E Klerk de (879_CR36) 2007; 109 S Burer (879_CR16) 2012; 40 N Gvozdenović (879_CR26) 2008; 19 K Gatermann (879_CR23) 2004; 192 E Klerk de (879_CR30) 2006; 20 K Murota (879_CR44) 2010; 27 E Klerk de (879_CR31) 2011; 159 S Bundfuss (879_CR14) 2009; 20 IM Bomze (879_CR7) 2000; 18 A Schrijver (879_CR56) 1979; 25 D Gijswijt (879_CR24) 2006; 113 I Dukanovic (879_CR21) 2010; 121 N Gvozdenović (879_CR25) 2008; 19 J Peña (879_CR48) 2007; 18 HD Mittelmann (879_CR43) 2010; 19 R Sotirov (879_CR58) 2009; 24 IM Bomze (879_CR11) 2002; 24 E Klerk de (879_CR39) 2010; 122 879_CR46 Y-Q Bai (879_CR4) 2009; 10 K Natarajan (879_CR45) 2011; 59 J Peña (879_CR47) 2006; 16 E Klerk de (879_CR34) 2006; 361 879_CR42 IM Bomze (879_CR6) 2012; 216 DR Woodall (879_CR63) 1993; 17 E Klerk de (879_CR29) 2010; 201 IM Bomze (879_CR9) 2010; 4 M Laurent (879_CR41) 2007; 109 879_CR49 J Povh (879_CR52) 2009; 6 E Klerk de (879_CR32) 2011; 129 J Sponsel (879_CR59) 2012; 52 F Vallentin (879_CR62) 2009; 430 J Lasserre (879_CR40) 2001; 11 879_CR2 K Zarankiewicz (879_CR64) 1954; 41 879_CR1
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Snippet	Copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization. An important line of... Issue Title: Special Issue: Polynomial Optimization Copositive programming is a relative young field which has evolved into a highly active research area in...
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SubjectTerms	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
Title	Exploiting symmetry in copositive programs via semidefinite hierarchies
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