We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that the use of SDP and RLT constraints together can produce bounds that are substantially better than either technique used alone. For highly symmetric problems we also consider the effect of symmetry-breaking based on tightened bounds on variables and/or order constraints.

The classical trust-region subproblem (TRS) minimizes a nonconvex quadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone constraints. For the case where an additional ellipsoidal constraint is added to TRS, resulting in the "two trust-region subproblem" (TTRS), we provide a new relaxation including second-order-cone constraints that strengthens the usual SDP relaxation.

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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,