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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.
On the setsemidefinite representation of nonconvex quadratic programs over arbitrary feasible sets
, 2011
"... ..."
Copositivity and Constrained Fractional Quadratic Problems
, 2011
"... ... formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branchandbound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and ..."
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Cited by 2 (2 self)
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... formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branchandbound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed.
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.
Preprint. Published in Journal of Global Optimization. DOI: 10.1007/s1089801300407 On the Exhaustivity of Simplicial Partitioning
, 2013
"... Abstract During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on t ..."
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Abstract During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counterexamples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity.
Copositivity tests based on the Linear Complementarity Problem
, 2014
"... Copositivity tests are presented based on new necessary and sufficient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke’s method, an enumerative algorithm and a linear mixedinteger programming formulation are proposed to solve the required L ..."
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Copositivity tests are presented based on new necessary and sufficient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke’s method, an enumerative algorithm and a linear mixedinteger programming formulation are proposed to solve the required LCPs. A new necessary condition for (strict) copositivity based on solving a Linear Program (LP) is also discussed, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to maxclique instances with matrices up to dimension 496 × 496. We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere.
INDEFINITE COPOSITIVE MATRICES WITH EXACTLY ONE POSITIVE EIGENVALUE OR EXACTLY ONE NEGATIVE EIGENVALUE
 ELA
, 2013
"... Checking copositivity of a matrix is a coNPcomplete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out ..."
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Checking copositivity of a matrix is a coNPcomplete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out if a matrix with exactly one negative eigenvalue is strictly copositive or not can be formulated as a combination of two convex quadratic programming problems which can be solved efficiently.
On the Exhaustivity of Simplicial Partitioning
"... Abstract During the last 40 years, simplicial partitioning has shown itself to be highly useful, including in the field of Nonlinear Optimisation. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at ..."
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Abstract During the last 40 years, simplicial partitioning has shown itself to be highly useful, including in the field of Nonlinear Optimisation. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity. Mathematics Subject Classification: 65K99; 90C26