On Lagrangian Relaxation of Quadratic Matrix Constraints

Abstract

Abstract. Quadratically constrained quadratic programs (QQPs) 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 assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the max-cut problem.Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, max-cut problems AMS subject classifications. 49M40, 52A41, 90C20, 90C27 PII. S0895479898340299 1. Introduction. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. They often provide a much improved model compared to the simpler linear relaxation of a problem. However, very large linear models can be solved efficiently, whereas QQPs are in general NP-hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Moreover these relaxations can be shown to be equivalent to semidefinite programming (SDP) relaxations, and SDP problems can be solved efficiently, i.e., they are polynomial time problems; see, e.g., [31].SDP relaxations provide a tractable approach for finding good bounds for many hard combinatorial problems. The best example is the application of SDP to the max-cut problem, where a 87% performance guarantee exists [11,12]. Other examples include matrix completion problems [23,22], as well as graph partitioning problems and the quadratic assignment problem (references given below).In this paper we consider several quadratically constrained quadratic (nonconvex) programs arising from hard combinatorial problems. In particular, we look at the orthogonal relaxations of the quadratic assignment and graph partitioning problems. We show that the resulting well-known eigenvalue bounds for these problems can be obtained from the Lagrangian dual of the orthogonally constrained relaxations,

The min-cut and vertex separator problem

“…We briefly describe this connection and then we consider more general models based on semidefinite optimization. The key tool here is the following theorem of Anstreicher and Wolkowicz [15], which can be viewed as an extension of the Hoffman-Wielandt theorem.…”

“…A quadratic programming (QP) relaxation for the min-cut problem is derived in [14]. That convex QP relaxation is based on the QP relaxation for the QAP, see [15,25,26]. Numerical results in [14] show that QP bounds are weaker, but cheaper to compute than the strongest SDP bounds, see also Sect.…”

“…by multiplying individual constraints from (15) and y ≥ 0. Finally we get in a similar way by multiplying with e − y ≥ 0…”

An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

“…Let (Y, y) be feasible for GPP ZWN and of block form (2). We construct from Y a feasible point Y ∈ S n for GPP RS in the following way:…”

“…Alizadeh [1] proved that the Donath-Hoffman bound is the dual of a semidefinite program (SDP). Also, Anstreicher and Wolkowicz [2] showed that the Donath-Hoffman bound can be obtained using the Lagrangian dual of an appropriate quadratically constrained problem.…”