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

Mathematical Programming deals with optimization programs of the form and includes the following general areas: minimize f(x) subject to gi(x) ≤ 0, i = 1,..., m, [x ⊂ R n] 1. Modelling: methodologies for posing various applied problems as optimization programs; 2. Optimization Theory, focusing on existence, uniqueness and on characterization of optimal solutions to optimization programs; 3. Optimization Methods: development and analysis of computational algorithms for various classes of optimization programs; 4. Implementation, testing and application of modelling methodologies and computational algorithms. Essentially, Mathematical Programming was born in 1948, when George Dantzig has invented Linear Programming – the class of optimization programs (P) with linear objective f(·) and

. We demonstrate that if A 1 ; :::; Am are symmetric positive semidefinite n \Theta n matrices with positive definite sum and A is an arbitrary symmetric n \Theta n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation max X fTr(AX) j Tr(A i X) 1; i = 1; :::; m; X 0g (SDP) of the optimization program x T Ax ! max j x T A i x 1; i = 1; :::; m (P) is not worse than 1 \Gamma 1 2 ln(2m 2 ) . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a feasible solution x to (P) with x T Ax Opt(SDP) 2 ln(2m 2 ) () can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all A i are of rank 1. Keywords: Semidefinite relaxations, quadratic programming 1. Introduction Let A i , i = 1; :::; m, be positive semidefinite n \Theta n matrices with positive definite sum, and A be a n \Theta n symmetric matrix. Con...

In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is known as the extended trust region subproblem and the computational complexity of this problem is still unknown. We consider several interesting cases related to this problem and show that for those cases the corresponding SDP relaxation admits no gap with the true optimal value, and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory which will lead to an optimal solution. Combining with a result obtained in the first part of the paper, we propose a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.

This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/11 − ɛ for all ɛ> 0. We show that it is quasi-NP-hard to approximate to a factor better than O(log γ n) for some γ> 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be Θ(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is log n Ω ( log log n), essentially answering one of the open problems of Alon et al. [AMMN]. 1.

The radius of a k-dimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The k-at radius of P , which we denote by R k (P ), is the minimum, over all k-dimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P .

Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced ...

We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex non-homogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating certain problems, such as quadratic optimization over the assignment polytope, according to the best of our knowledge.

In this paper we study a class of quadratic maximization problems and their semidefinite programming (SDP) relaxation. For a special subclass of the problems we show that the SDP relaxation provides an exact optimal solution. Another subclass, which is NP-hard, guarantees that the SDP relaxation yields an approximate solution with a worst-case performance ratio of 0:87856:::. This is a generalization of the well-known result of Goemans and Williamson for the maximum-cut problem. Finally, we discuss extensions of these results in the presence of a certain type of sign restrictions. Key words: Quadratic programming, semidefinite programming relaxation, polynomial-time solvability, approximation. AMS subject classification: 90C20, 90C26. 1 Introduction Semidefinite programming has been an active research area following the seminal work of Nesterov and Nemirovski [7]. It has wide applications in many directions including engineering, economics and combinatorial optimization. In the latte...

We consider the problem of computing the outer-radii of point sets. Inthis problem, we are given integers n, d, k where k < = d, and a set P of n points in Rd. The goal is to compute the outer k-radius of P, denoted by Rk(P), which is the minimum, over all (d- k)-dimensional flats F, ofmax p2P d(p, F), where d(p, F) is the Euclidean distance between the point p and flat F. Computing the radii of point sets is a fundamental problem incomputational convexity with significantly many applications. The problem admits a polynomial time algorithm when the dimension d is constant [15].Here we are interested in the general case when the dimension d is not fixedand can be as large as n, where the problem becomes NP-hard even for k = 1.It is known that Rk(P) can be approximated in polynomial time by afactor of (1 + "), for any "> 0, when d- k is a fixed constant [22, 7]. A