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COMPUTING SEMIDEFINITE PROGRAMMING LOWER BOUNDS FOR THE (FRACTIONAL) CHROMATIC NUMBER VIA . . .
 SIAM J. OPTIM. VOL. 19, NO. 2, PP. 592–615
, 2008
"... Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572–591] hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. In particular, we introduced two hierarchies of lower bounds: the “ψ”hierarchy converging to the fractional chromatic number and the “Ψ”hierar ..."
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Cited by 6 (1 self)
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Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572–591] hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. In particular, we introduced two hierarchies of lower bounds: the “ψ”hierarchy converging to the fractional chromatic number and the “Ψ”hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lovász theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed in [SIAM J. Optim., 19 (2008), pp. 572–591]. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs, and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits us to compute the bounds by using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit blockdiagonalization of the Terwilliger algebra given by Schrijver [IEEE Trans. Inform. Theory, 51 (2005), pp. 2859–2866]. Our numerical results indicate that the new bounds can be much stronger than the Lovász theta number. For some of the DIMACS instances we improve the best known lower bounds significantly.
Blockdiagonal semidefinite programming hierarchies for 0/1 programming
 OPERATIONS RESEARCH LETTERS
, 2009
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Invariant semidefinite programs
 IN HANDBOOK ON SEMIDEFINITE, CONIC AND POLYNOMIAL OPTIMIZATION (M.F. ANJOS
, 2012
"... This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization. ..."
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Cited by 4 (2 self)
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This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.
Copositive optimization – recent developments and applications
 European Journal of Operational Research
, 2012
"... Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear const ..."
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Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NPhard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications. 1.
Convex Relaxations and Integrality Gaps
"... Summary. We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increa ..."
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Summary. We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems. 1
Team RealOpt Reformulation based algorithms for Combinatorial Optimization
"... c t i v it y e p o r t 2007 Table of contents ..."
Team RealOpt Reformulation based algorithms for Combinatorial Optimization
"... c t i v it y e p o r t 2008 Table of contents ..."
On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual
, 2012
"... Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi [K.G. Murty and S.N. Kabadi, Some N Pcomplete problems in quadratic and nonlinear programming, Mathematical Programming, 39, no.2:117–129, 1987] that the ..."
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Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi [K.G. Murty and S.N. Kabadi, Some N Pcomplete problems in quadratic and nonlinear programming, Mathematical Programming, 39, no.2:117–129, 1987] that the strong membership problem for the copositive cone, that is deciding whether or not a given matrix is in the copositive cone, is a coN Pcomplete problem. From this it has long been assumed that this implies that the question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an N Phard problem. However, the technical details for this have not previously been looked at to confirm that this is true. In this paper it is proven that the strong membership problem for the completely positive cone is indeed N Phard. Furthermore, it is shown that even the weak membership problems for both of these cones are N Phard. We also present an alternative proof of the N Phardness of the strong membership problem for the copositive cone. 1
COMPUTABLE REPRESENTATION OF THE CONE OF NONNEGATIVE QUADRATIC FORMS OVER A GENERAL SECONDORDER CONE AND ITS APPLICATION TO COMPLETELY POSITIVE PROGRAMMING
"... Abstract. In this paper, we provide a computable representation of the cone of nonnegative quadratic forms over a general nontrivial secondorder cone using linear matrix inequalities (LMI). By constructing a sequence of such computable cones over a union of secondorder cones, an efficient algorith ..."
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Abstract. In this paper, we provide a computable representation of the cone of nonnegative quadratic forms over a general nontrivial secondorder cone using linear matrix inequalities (LMI). By constructing a sequence of such computable cones over a union of secondorder cones, an efficient algorithm is designed to find an approximate solution to a completely positive programming problem using semidefinite programming techniques. In order to accelerate the convergence of the approximation sequence, an adaptive scheme is adopted, and “reformulationlinearization technique ” (RLT) constraints are added to further improve its efficiency. 1. Introduction. Studying