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Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 106 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 89 (3 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Solving Large Quadratic Assignment Problems on Computational Grids
, 2000
"... The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computat ..."
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Cited by 82 (7 self)
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The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a stateoftheart branchandbound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported.
Optimal Downlink Beamforming Using Semidefinite Optimization
, 1999
"... When using antenna arrays at the base station of a cellular system, one critical aspect is the transmission strategy. An optimal choice of beamformers for simultaneous transmission to several cochannel users must be solved jointly for all users and base stations in an area. We formulate an optimal ..."
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Cited by 58 (5 self)
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When using antenna arrays at the base station of a cellular system, one critical aspect is the transmission strategy. An optimal choice of beamformers for simultaneous transmission to several cochannel users must be solved jointly for all users and base stations in an area. We formulate an optimal transmit strategy and show how the solution can be calculated efficiently using interior point methods for semidefinite optimization. The algorithm minimizes the total transmitted power under certain constraints to guarantee a specific quality of service. The method provides large flexibility in the choice of constraints and can be extended to be robust to channel perturbations.
Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints
 SIAM JOURNAL ON OPTIMIZATION
, 2006
"... We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection betwe ..."
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Cited by 44 (9 self)
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We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.
A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming
 MATHEMATICAL PROGRAMMING
, 1999
"... We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be comp ..."
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Cited by 37 (4 self)
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We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort.
Copositive and semidefinite relaxations of the quadratic assignment problem
 DISCRETE OPTIM
, 2009
"... Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the co ..."
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Cited by 33 (3 self)
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Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the wellstudied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.
SEMIDEFINITE PROGRAMMING RELAXATIONS FOR THE GRAPH PARTITIONING PROBLEM
, 1999
"... A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 co ..."
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Cited by 31 (6 self)
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A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 constraints in GP. The special structure of the relaxation is exploited in order to project onto the minimal face of the cone of positive semidefinite matrices which contains the feasible set. This guarantees that the Slater constraint qualification holds, which allows for a numerically stable primaldual interiorpoint solution technique. A gangster operator is the key to providing an efficient representation of the constraints in the relaxation. An incomplete preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. Only dual feasibility is enforced, which results in the desired lower bounds, but avoids the expensive primal feasibility calculations. Numerical results
Sums of random symmetric matrices and quadratic optimization under orthogonality constraints
 MATHEMATICAL PROGRAMMING (2006), ONLINE FIRST ISSUE, DOI
"... Let Bi be deterministic real symmetric m × m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested to know under what conditions “typical norm ” of the random matrix SN = N� ξiBi is of order of 1. An evident necessary condition ..."
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Cited by 30 (3 self)
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Let Bi be deterministic real symmetric m × m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested to know under what conditions “typical norm ” of the random matrix SN = N� ξiBi is of order of 1. An evident necessary condition is E{S 2 N to N� B i=1 2 i i=1} � O(1)I, which, essentially, translates � I; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of SN is ≤ O(1)m 1 6: Prob{�SN �> Ωm 1/6} ≤ O(1) exp{−O(1)Ω 2} for all Ω> 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints Opt = max Xj∈Rm×m � F (X1,..., Xk) : XjX T j = I, j = 1,..., k �, where F is quadratic in X = (X1,..., Xk). We show that when F is convex in every one of Xj, a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices Xj: Opt ≤ Opt(SDP) ≤ O(1)[m1/3 + ln k]Opt.