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41
Convex analysis on the Hermitian matrices
 SIAM Journal on Optimization
, 1996
"... There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions ..."
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Cited by 45 (20 self)
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There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues. A new approach to this characterization is given, via a simple Fenchel conjugacy formula. We then apply this formula to derive expressions for subdifferentials, and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables. Analogous results hold for Hermitian matrices. Key Words: convexity, matrix function, Schur convexity, Fenchel duality, subdifferential, unitarily invariant, spectral function, positive semidefinite programming, quasiNewton update. AMS 1991 Subject Classification: Primary 15A45 49N15 Secondary 90C25 65K10 1 Introduction A matrix norm on the n \Theta n complex matrices is called unitarily inv...
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 42 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
A Projection Technique for Partitioning the Nodes of a Graph
, 1995
"... Let G = (N; E) be a given undirected graph. We present several new techniques for partitioning the node set N into k disjoint subsets of specified sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these ..."
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Cited by 41 (13 self)
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Let G = (N; E) be a given undirected graph. We present several new techniques for partitioning the node set N into k disjoint subsets of specified sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these techniques to be very successful. 1 Introduction Let G = (N; E) be a given undirected graph with node set N = f1; : : : ; ng and edge set E. A common problem in circuit board and microchip design, computer program segmentation, floor planning and other layout problems is to partition the node set N into k disjoint subsets S 1 ; : : : ; S k of specified sizes m 1 m 2 : : : m k ; P k j=1 m j = n, so as to minimize the number of edges connecting nodes in distinct subsets of the partition. We refer to an edge, which connects nodes in distinct subsets of the partition, as being cut by the partition. A recent survey on the graph partitioning problem and further related problems is containe...
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.
A Hypergraph Framework For Optimal ModelBased Decomposition Of Design Problems
 Computational Optimization and Applications
, 1997
"... Decomposition of large engineering system models is desirable since increased model size reduces reliability and speed of numerical solution algorithms. The article presents a methodology for optimal modelbased decomposition (OMBD) of design problems, whether or not initially cast as optimization p ..."
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Cited by 30 (20 self)
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Decomposition of large engineering system models is desirable since increased model size reduces reliability and speed of numerical solution algorithms. The article presents a methodology for optimal modelbased decomposition (OMBD) of design problems, whether or not initially cast as optimization problems. The overall model is represented by a hypergraph and is optimally partitioned into weakly connected subgraphs that satisfy decomposition constraints. Spectral graphpartitioning methods together with iterative improvement techniques are proposed for hypergraph partitioning. A known spectral Kpartitioning formulation, which accounts for partition sizes and edge weights, is extended to graphs with also vertex weights. The OMBD formulation is robust enough to account for computational demands and resources and strength of interdependencies between the computational modules contained in the model. KEYWORDS: Model decomposition, multidisciplinary design, hypergraph partitioning, larges...
Solving Large Quadratic Assignment Problems in Parallel.
 Computational Optimization and Applications
, 1994
"... . Quadratic Assignment problems are in practice among the most difficult to solve in the class of NPcomplete problems. The only successful approach hitherto has been BranchandBound based algorithms, but such algorithms are crucially dependent on good bound functions to limit the size of the space ..."
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Cited by 24 (6 self)
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. Quadratic Assignment problems are in practice among the most difficult to solve in the class of NPcomplete problems. The only successful approach hitherto has been BranchandBound based algorithms, but such algorithms are crucially dependent on good bound functions to limit the size of the space searched. Much work has been done to identify such functions for the QAP, but with limited success. Parallel processing has also been used in order to increase the size of problems solvable to optimality. The systems used have, however, often been systems with relatively few, but very powerful vector processors, and have hence not been ideally suited for computations essentially involving nonvectorizable computations on integers. In this paper we investigate the combination of one of the best bound functions for a Branchand Bound algorithm (the GilmoreLawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in a parallel BranchandBo...
Selected Topics on Assignment Problems
, 1999
"... We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and co ..."
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Cited by 22 (1 self)
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We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems.
Lower bounds for the quadratic assignment problem
 University of Munich
, 1994
"... Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new ..."
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Cited by 20 (5 self)
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Abstract. We investigate the classical GilmoreLawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the GilmoreLawler Bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branchandbound type algorithm for the quadratic assignment problem. 1.
Convex Relaxations Of 01 Quadratic Programming
, 1993
"... We consider three parametric relaxations of the 01 quadratic programming problem. These relaxations are to: quadratic maximization over simple box constraints, quadratic maximization over the sphere, and the maximum eigenvalue of a bordered matrix. When minimized over the parameter, each of the rel ..."
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Cited by 17 (6 self)
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We consider three parametric relaxations of the 01 quadratic programming problem. These relaxations are to: quadratic maximization over simple box constraints, quadratic maximization over the sphere, and the maximum eigenvalue of a bordered matrix. When minimized over the parameter, each of the relaxations provides an upper bound on the original discrete problem. Moreover, these bounds are efficiently computable. Our main result is that, surprisingly, all three bounds are equal. This author would like to thank the Department of Civil Engineering and Operations Research, Princeton University, for their support during his research leave. Key words: quadratic boolean programming, bounds, quadratic programming, trust region subproblems, minmax eigenvalue problems. AMS 1991 Subject Classification: Primary: 90C09, 90C25; Secondary: 90C27, 90C20. 1 INTRODUCTION Consider the \Sigma1 quadratic programming problem (P ) ¯ := max q(x) := x t Qx + c t x; x 2 F := f\Gamma1; 1g n ; ...
Bounds for the Quadratic Assignment Problem Using Continuous Optimization Techniques
, 1990
"... The quadratic assignment problem (denoted QAP ), in the trace formulation over the permutation matrices, is min X2\Pi tr(AXB +C)X t : Several recent lower bounds for QAP are discussed. These bounds are obtained by applying continuous optimization techniques to approximations of this combinator ..."
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Cited by 16 (5 self)
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The quadratic assignment problem (denoted QAP ), in the trace formulation over the permutation matrices, is min X2\Pi tr(AXB +C)X t : Several recent lower bounds for QAP are discussed. These bounds are obtained by applying continuous optimization techniques to approximations of this combinatorial optimization problem, as well as by exploiting the special matrix structure of the problem. In particular, we apply constrained eigenvalue techniques, reduced gradient methods, subdifferential calculus, generalizations of trust region methods, and sequential quadratic programming. Keywords : Quadratic Assignment Problem, Bounds, Constrained Eigenvalues, Reduced Gradient, Trust Regions, Sequential Quadratic Programming. 1 Introduction The quadratic assignment problem, denoted QAP , is a generalization of the linear sum assignment problem, i.e. given the set N = f1; 2; : : : ; ng and three n by n matrices A = (a ik ); B = (B jl ); and C = (c ij ); find a permutation ß of the set N which m...