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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.

Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y

We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation Q t (G) can be defined as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of [1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I ; J)-th entry depends only on the symmetric difference of the sets I and J . The set F 0 (n) is the basic semidefinite relaxation of max-cut consisting of the semidefinite matrices of order n with an all ones diagonal, while Fn\Gamma2 (n) is the 2 n\Gamma1 -dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n \Gamma 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively. 1

The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primal-dual interior-point technique which uses an inexact Gauss-Newton approach with a matrix free preconditioned conjugate gradient method. This approach avoids the ill-conditioning pitfalls that result from symmetrization and from forming the so-called normal equations, while maintaining the primal-dual framework.

The Quadratic Assignment Problem, QAP, is arguably the hardest of the NP-hard problems. One of the main reasons is that it is very difficult to get good quality bounds for branch and bound algorithms. We show that many of the bounds that have appeared in the literature can be ranked and put into a unified Semidefinite Programming, SDP, framework. This is done using redundant quadratic constraints and Lagrangian relaxation. Thus, the final SDP relaxation ends up being the strongest.

. Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interior-point methods. In this paper we study these semidefinite relaxations using the equivalent Lagrangian relaxations. In particular, the theme of the paper is to show that the Lagrangian relaxation is, in some respects, best. In all instances we consider, we show that whenever we have a tractable bound (relaxation), then the same bound can be obtained from a Lagrangian relaxation. Keywords: Semidefinite Programming, Lagrangian Duality, Relaxations, Quadratic Constrained Quadratic Programs, Hard combinatorial Problems. Table of Contents 1 Introduction 2 1.1 Lagrange Multipliers for Q 2 P 3 1.2 Semidefinite Programming Preliminaries 5 2 Relaxations of Q 2 P 8 2.1 Relaxations for the Max-cut Problem 8 2.2 General Q 2 P 11 2.3...

About these Notes: Semidefinite Programming, SDP, refers to optimization problems where the vector variable is a symmetric matrix which is required to be positive semidefinite. Though SDPs (under various names) have been studied as far back as the 1940s, the interest has grown tremendously during the last ten years. This is partly due to the many diverse applications in e.g. engineering, combinatorial optimization, and statistics. Part of the interest is due to the great advances in efficient solutions for these types of problems. These notes summarize the theory, algorithms, and applications for semidefinite programming. They were prepared for a minicourse given at the Workshop on Large Scale Nonlinear and Semidefinite

In this paper we summarize recent results on finding tight semidefinite programming relaxations for the Max-Cut problem and hence tight upper bounds on its optimal value. Our results hold for every instance of Max-Cut and in particular we make no assumptions on the edge weights. We present two strengthenings of the well-known semidefinite programming relaxation of Max-Cut studied by Goemans and Williamson. Preliminary numerical results comparing the relaxations on several interesting instances of Max-Cut are also presented.