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.tu-graz.ac.at/qaplib/

We present a dual-scaling interior-point 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 interior-point algorithms for approximating the maximum cut semidefinite programs with dimension up-to 3000.

We describe a few applications of semide nite programming in combinatorial optimization.

Abstract. The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and the Nesterov-Todd search directions have been used in many primal-dual interior-point 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.

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 state-of-the-art branch-and-bound 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.

Abstract not found

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 interior-point 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.

Quadratically constrained quadratic programs (QQP) 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 XX T = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XX T = I, and the seemingly redundant constraints X T X = I, has a zero duality gap. This result has natural applications to quadratic assignm...

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...

Most current implementations of interior-point methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) M y = h in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided if an iterative method is used instead. Numerical experiments have shown that the conjugate residual (CR) method typically takes a huge number of steps to generate a high accuracy solution. On the other hand, it is difficult to incorporate traditional preconditioners into the SCE, except for block diagonal preconditioners. We decompose the SCE into a 2 × 2 block system by decomposing y (similarly for h) into two orthogonal components with one lying in a certain subspace that is determined from the structure of M . Numerical experiments on semidefinite programming problems arising from Lovász -function of graphs and MAXCUT problems show that high accuracy solutions can be obtained with moderate n...