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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.
An Adaptive Linear Approximation Algorithm for Copositive Programs
, 2008
"... We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization; for instance, the maximum clique problem and other combinatorial problems can be reformulated as such problems. We present new polyhedral inner and outer ..."
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We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization; for instance, the maximum clique problem and other combinatorial problems can be reformulated as such problems. We present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit. In contrast to previous approximation schemes, our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function, yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not. Using these approximations, we derive an adaptive linear approximation algorithm for copositive programs. Numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems.
AN IMPROVED CHARACTERISATION OF THE INTERIOR OF THE COMPLETELY POSITIVE CONE
, 2010
"... A symmetric matrix is defined to be completely positive if it allows a factorisation BB T, where B is an entrywise nonnegative matrix. This set is useful in certain optimisation problems. The interior of the completely positive cone has previously been characterised by Dür and Still [M. Dür and G. ..."
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A symmetric matrix is defined to be completely positive if it allows a factorisation BB T, where B is an entrywise nonnegative matrix. This set is useful in certain optimisation problems. The interior of the completely positive cone has previously been characterised by Dür and Still [M. Dür and G. Still, Interior points of the completely positive cone, Electronic Journal of Linear Algebra, 17:48–53, 2008]. In this paper, we introduce the concept of the set of zeros in the nonnegative orthant for a quadratic form, and use the properties of this set to give a more relaxed characterisation of the interior of the completely positive cone.
On the accuracy of uniform polyhedral approximations of the copositive cone
, 2009
"... 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 approximatio ..."
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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 closedform 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.
Mixed zeroone linear programs under objective uncertainty: a completely positive representation
, 2009
"... In this paper, we analyze mixed 01 linear programs under objective uncertainty. The mean vector and the second moment matrix of the nonnegative objective coefficients is assumed to be known, but the exact form of the distribution is unknown. Our main result shows that computing a tight upper bound ..."
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In this paper, we analyze mixed 01 linear programs under objective uncertainty. The mean vector and the second moment matrix of the nonnegative objective coefficients is assumed to be known, but the exact form of the distribution is unknown. Our main result shows that computing a tight upper bound on the expected value of a mixed 01 linear program in maximization form with random objective is a completely positive program. This naturally leads to semidefinite programming relaxations that are solvable in polynomial time but provide weaker bounds. The result can be extended to deal with uncertainty in the moments and more complicated objective functions. Examples from order statistics and project networks highlight the applications of the model. Our belief is that the model will open an interesting direction for future research in discrete and linear optimization under uncertainty.
On the computation of C ∗ certificates
, 2008
"... The cone of completely positive matrices C ∗ is the convex hull of all symmetric rank1matrices xx T with nonnegative entries. While there exist simple certificates proving that a given matrix B ∈ C ∗ is completely positive it is a rather difficult problem to find such a certificate. We examine a s ..."
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The cone of completely positive matrices C ∗ is the convex hull of all symmetric rank1matrices xx T with nonnegative entries. While there exist simple certificates proving that a given matrix B ∈ C ∗ is completely positive it is a rather difficult problem to find such a certificate. We examine a simple algorithm which – for a given input B – either determines a certificate proving that B ∈ C ∗ or converges to a matrix ¯ S in C ∗ which in some sense is “close ” to B. Numerical experiments on matrices B of dimension up to 200 conclude the presentation. Key words. Completely positive matrices. 1
Separation and relaxation for cones of quadratic forms
, 2011
"... Let P ⊆ ℜn be a pointed, polyhedral cone. In this paper, we study the cone C = cone{xxT: x ∈ P} of quadratic forms. Understanding the structure of C is important for globally solving NPhard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C p ..."
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Let P ⊆ ℜn be a pointed, polyhedral cone. In this paper, we study the cone C = cone{xxT: x ∈ P} of quadratic forms. Understanding the structure of C is important for globally solving NPhard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C provided one can optimize with respect to a related cone over the boundary of P. This algorithm leads to a nonlinear representation of C and a class of tractable relaxations for C, each of which improves a standard polyhedralsemidefinite relaxation of C. The relaxation technique can further be applied recursively to obtain a hierarchy of relaxations, and for constant recursive depth, the hierarchy is tractable. We apply this theory to two important cases: P is the nonnegative orthant, in which case C is the cone of completely positive matrices; and P is the homogenized cone of the “box ” [0, 1] n. Through various results and examples, we demonstrate the strength of the theory for these cases. For example, we achieve for the first time a separation algorithm for 5 × 5 completely positive matrices.
A LagrangianDNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems
, 2013
"... We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and firstorder algorithms. The simplified LagrangianCPP relaxation of such QOPs proposed b ..."
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We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and firstorder algorithms. The simplified LagrangianCPP relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a completely positive (CPP) matrix variable with its upperleft element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the LagrangianDNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the LagrangianDNN relaxation. We then propose an efficient numerical method for the LagrangianDNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.
Building a completely positive factorization
"... Using a bordering approach, and building upon an already known factorization of a principal block, we establish sufficient conditions under which we can extend this factorization to the full matrix. Simulations show that the approach is promising also in higher dimensions, provided there is no gap ..."
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Using a bordering approach, and building upon an already known factorization of a principal block, we establish sufficient conditions under which we can extend this factorization to the full matrix. Simulations show that the approach is promising also in higher dimensions, provided there is no gap in the cprank. We expect that this property is shared by good quality approximation solutions obtained by the usual conic (semidefinite) relaxation procedures in copositive programming for combinatorial optimization applications. Key words. Copositive programming, semidefinite relaxation, mixedbinary quadratic optimization 1
ON THE IRREDUCIBILITY, SELFDUALITY, AND NONHOMOGENEITY OF COMPLETELY POSITIVE CONES
, 2013
"... For a closed cone C in R n, the completely positive cone of C is the convex cone KC in S n generated by {uu T: u ∈ C}. Such a cone arises, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by ..."
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For a closed cone C in R n, the completely positive cone of C is the convex cone KC in S n generated by {uu T: u ∈ C}. Such a cone arises, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by the useful and desirable properties of the nonnegative orthant and the positive semidefinite cone (and more generally of symmetric cones in Euclidean Jordan algebras), this paper investigates when (or whether) KC can be irreducible, selfdual, or homogeneous.