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62
QAPLIB  A Quadratic Assignment Problem Library
, 1996
"... This report, the data and also most of the best feasible solutions are available via World Wide Web. The URLs of the QAPLIB Home Page are http://www.opt.math.tugraz.ac.at/qaplib/ ..."
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Cited by 163 (6 self)
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This report, the data and also most of the best feasible solutions are available via World Wide Web. The URLs of the QAPLIB Home Page are http://www.opt.math.tugraz.ac.at/qaplib/
Solving LargeScale Sparse Semidefinite Programs for Combinatorial Optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational re ..."
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Cited by 116 (11 self)
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We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interiorpoint algorithms for approximating the maximum cut semidefinite programs with dimension upto 3000.
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 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Exploiting Sparsity in PrimalDual InteriorPoint Methods for Semidefinite Programming
 Mathematical Programming
, 1997
"... Abstract. The HelmbergRendlVanderbeiWolkowicz/KojimaShindohHara/Monteiro and the NesterovTodd search directions have been used in many primaldual interiorpoint methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite prog ..."
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Cited by 68 (21 self)
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Abstract. The HelmbergRendlVanderbeiWolkowicz/KojimaShindohHara/Monteiro and the NesterovTodd search directions have been used in many primaldual interiorpoint methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite program to be solved is large scale and sparse.
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 68 (6 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.
On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 50 (19 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
Strong duality for semidefinite programming
 SIAM J. Optim
, 1997
"... Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite ..."
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Cited by 47 (19 self)
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Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.
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 33 (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.
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 27 (8 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.