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 micro-chip 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...

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, quasi-Newton 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...

A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X , 1 (X) 2 (X) : : : n (X), and hence may be written f( 1 (X); 2 (X); : : : ; n (X)) for some symmetric function f . Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is differentiable at X if and only if the function f is differentiable at the vector (X), and we give a concise formula for the derivative. We then apply this formula to deduce an analogous expression for the Clarke generalized gradient of the spectral function. A similar result holds for real symmetric matrices. 1 Introduction and notation Optimization problems involving a symmetric matrix variable, X say, frequently involve symmetric functions of the eigenvalues of X in the objective or constraints. Examples include the maximum eigenvalue of X, or log(det X) (for positive definite X), or eigenvalue constraints such as positive semidefinit...

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 well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort. Keywords: Quadratic Assignment Problem, Eigenvalue Bounds, Quadratic Programming, Semidefinite Programming. Dept. of Management Sciences, University of Iowa, Iowa City, IA 52242 y Dept. of Computer Science, University of Iowa, Iowa City, IA 52242 1 Introduction The quadratic assignment problem (QAP) in "Koopmans-Beckmann" form can be written QAP(A;B;C) : min tr(AXB + C)X T s:t: X 2 \Pi; where A, B and C are n \Theta n matrices, tr denotes the trace of a matrix, and \Pi is the set of n \Theta n permutation matrices. Throughout we assume that A and B are symmetric. The QAP is a very well-know...

. Quadratic Assignment problems are in practice among the most difficult to solve in the class of NP-complete problems. The only successful approach hitherto has been Branch-andBound -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 non-vectorizable computations on integers. In this paper we investigate the combination of one of the best bound functions for a Branchand -Bound algorithm (the Gilmore-Lawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in a parallel Branch-and-Bo...

. We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler 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 branch-andbound type algorithm for the quadratic assignment problem. Key words. quadratic assignment problem, branch-and-bound, lower bound, combinatorial optimization AMS(MOS) subject classifications. AMS(MOS) 68Q25, 90B80, 90C27. 1. Introduction. We consider the quadratic assignment problem (QAP) in the Koopmans and Beckmann form [18]. Given a positive integer n and two n \Theta n matrices A = (a ij ) and B = (b ij ), the problem is to find a permutation p of the set f1; 2; : : :; ng that minimizes n X i=1 n X j=1 a ij b p(i)p(j): The QAP belongs to a class of combinatorial optimization problems with many practical a...

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 model-based 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 graph-partitioning methods together with iterative improvement techniques are proposed for hypergraph partitioning. A known spectral K-partitioning 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...

We consider three parametric relaxations of the 0-1 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 ; ...

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...

Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP -hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the so-called "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primal-dual inter...