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33
On the copositive representation of binary and continuous nonconvex quadratic programs
, 2007
"... In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of ..."
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Cited by 89 (6 self)
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In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of a small collection of NPhard problems. A simplification, which reduces the dimension of the linear conic program, and an extension to complementarity constraints are established, and computational issues are discussed.
A NewtonCG augmented Lagrangian method for semidefinite programming
 SIAM J. Optim
"... Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresp ..."
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Cited by 63 (12 self)
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Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth NewtonCG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts.
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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Cited by 29 (0 self)
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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.
Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem
, 2009
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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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Cited by 11 (1 self)
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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.
Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices based on Semidefinite Programming
, 2008
"... Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the is ..."
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Cited by 10 (4 self)
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Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes of QAPs based on semidefinite programming (SDP). By exploiting the data structure of the distance matrix B, we first show that for any permutation matrix X, the matrix Y = αE − XBX T is positive semidefinite, where α is a properly chosen parameter depending only on the associated graph in the underlying QAP and E = ee T is a rank one matrix whose elements have value 1. This results in a natural way to approximate the original QAPs via SDP relaxation based on the matrix splitting technique. Our new SDP relaxations have a smaller size compared with other SDP relaxations in the literature and can be solved efficiently by most open source SDP solvers. Experimental results show that for the underlying QAPs of size up to n=200, strong bounds can be obtained effectively.
On the setsemidefinite representation of nonconvex quadratic programs over arbitrary feasible sets
, 2011
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SDP relaxations for some combinatorial optimization problems
, 2010
"... In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known ..."
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Cited by 5 (4 self)
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In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem.