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53
Quadratic matrix programming
 SIAM J. Optim
"... We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidef ..."
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Cited by 27 (2 self)
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We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidefinite relaxation (SDR) and dual for the QMP problem and show that under some mild conditions strong duality holds for QMP problems with at most r constraints. Using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form, we are able to compare the constructed SDR and dual problems to other known SDR and dual formulations of the problem. An application to robust least squares problems is discussed. 1
Solving Quadratic Assignment Problems Using Convex Quadratic Programming Relaxations
, 2000
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Multivariate nonnegative quadratic mappings
 SIAM J. Optim
, 2002
"... Abstract. In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a prespecified conic order. In particular, we consider the set (cone) of nonnegative quadrat ..."
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Cited by 17 (7 self)
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Abstract. In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a prespecified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study when it can be represented by linear matrix inequalities. We also discuss the applications of the results in robust optimization, especially the robust quadratic matrix inequalities and the robust linear programming models. In the latter application the implementational errors of the solution is taken into account, and the problem is formulated as a semidefinite program. Key words. Linear matrix inequalities, convex cone, robust optimization, biquadratic functions AMS subject classifications. 15A48, 90C22
Approximating global quadratic optimization with convex quadratic constraints
, 1998
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A STRENGTHENED SDP RELAXATION via a SECOND LIFTING for the MAXCUT PROBLEM
, 1999
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A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid
"... the date of receipt and acceptance should be inserted later Abstract We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the socalled Regularized Total Least Squares problem (RTLS) ..."
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Cited by 15 (3 self)
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the date of receipt and acceptance should be inserted later Abstract We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the socalled Regularized Total Least Squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance ε grows as O ( √ ln ε −1). Keywords ratio of quadratic minimization · nonconvex quadratic minimization · semidefinite programming · strong duality · regularized total least squares · fixed point algorithms · convergence analysis
Strong Duality for a TrustRegion Type Relaxation of the Quadratic Assignment Problem
, 1998
"... Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic p ..."
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Cited by 15 (9 self)
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Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced ...
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Cited by 14 (5 self)
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for MetaHeuristic Methods
 Annals of OR
, 2005
"... This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP). QAP instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly foun ..."
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Cited by 14 (1 self)
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This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP). QAP instances frequently used in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristicbased methods. However, these new instances are shown to be solved relatively well by some exact methods, since problem instances up to a size of 75 have been exactly solved. Key words: Quadratic assignment problem, local search, branch & bound, benchmarks. 1. Introduction. 1.1 The quadratic assignment problem (QAP). The QAP is a combinatorial optimization problem stated for the first time by Koopmans and Beckmann in 1957. It can be described as follows: Given two n × n matrices (aij) and(bkl), find a permutation ππππ minimizing: