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Copositive Programming  a Survey
, 2009
"... Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial ..."
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Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sumofsquares approaches, as well as algorithmic solution approaches for copositive programs.
Convex Relaxations and Integrality Gaps
"... 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 st ..."
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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.
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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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.
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 7 (3 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.
Zeroerror sourcechannel coding with entanglement
, 2013
"... Abstract. We study the use of quantum entanglement in the zeroerror sourcechannel coding problem. Here, Alice and Bob are connected by a noisy classical oneway channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice’s input while using the channel as li ..."
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Abstract. We study the use of quantum entanglement in the zeroerror sourcechannel coding problem. Here, Alice and Bob are connected by a noisy classical oneway channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice’s input while using the channel as little as possible. In the zeroerror regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lovász theta number, a graph parameter defined by a semidefinite program, gives the best efficientlycomputable upper bound on the Shannon capacity and it also upper bounds its entanglementassisted counterpart. At the same time it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here we partially extend these results to the sourcecoding problem and to the more general sourcechannel coding problem. We prove a lower bound on the rate of entanglementassisted sourcecodes in terms Szegedy’s number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical sourcechannel codes. Our separation results use lowdegree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu and a new application of remote state preparation.
Blockdiagonal semidefinite programming hierarchies for 0/1 programming
 OPERATIONS RESEARCH LETTERS
, 2009
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Optimization over Polynomials: Selected Topics
"... Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of s ..."
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Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidefinite optimization. Sums of squares are used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies to some combinatorial graph problems, and discuss links with other relaxation methods.
COMPUTABLE REPRESENTATION OF THE CONE OF NONNEGATIVE QUADRATIC FORMS OVER A GENERAL SECONDORDER CONE AND ITS APPLICATION TO COMPLETELY POSITIVE PROGRAMMING
, 2013
"... 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 des ..."
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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.