Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Abstract

We study nonconvex quadratic minimization problems under (possibly nonconvex) quadratic and linear constraints, characterizing both Lagrangian and semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be novel), we show that the semi-Lagrangian dual is equivalent to a natural copositive relaxation (and this has apparently not been observed before). This way, we arrive at conic bounds tighter than the usual Lagrangian dual (and thus than the SDP) bounds. Any of the known tractable inner approximations of the copositive cone can be used for this tightening, but in particular, the above-mentioned characterization with explicit linear constraints is a natural, much cheaper, relaxation than the usual zero-order approximation by doubly nonnegative (DNN) matrices and still improves upon the Lagrangian dual bounds. These approximations are based on LMIs on matrices of basically the original order plus additional linear constraints (in contrast to more familiar sum-of-squares or moment approximation hierarchies) and thus may have merits in particular for large instances where it is important to employ only a few inequality constraints (e.g., n instead of n(n−1) 2 for the DNN relaxation). Further, we specify sufficient conditions for tightness of the semi-Lagrangian relaxation and show that copositivity of the slack matrix guarantees global optimality for KKT points of this problem, thus significantly improving upon a well-known secondorder global optimality condition.

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

“…A form of partial Lagrangian relaxation called semi-Lagrangian of (P) (see [7,17] and the references therein) is given by…”

“…Of course, this happens with higher frequency than positive-definiteness of the Hessian, and the discrepancy is not negligible, see [7,Section 5] for an example.…”

“…By now, equivalent copositive reformulations for many important problems are known, among them (non-convex, mixed-binary, fractional) quadratic optimization problems under a mild assumption [2,3,13], and some special optimization problems under uncertainty [4,18,32,37]. In particular, it has been shown in [7] that, for quadratic optimization problems with additional nonnegative constraints, copositive relaxations (and its tractable approximations) provides a tighter bound than the usual Lagrangian relaxation. On the other hand, the techniques in [7] are not directly applicable because our model problem does not require the variables to be nonnegative.…”

“…In particular, it has been shown in [7] that, for quadratic optimization problems with additional nonnegative constraints, copositive relaxations (and its tractable approximations) provides a tighter bound than the usual Lagrangian relaxation. On the other hand, the techniques in [7] are not directly applicable because our model problem does not require the variables to be nonnegative.…”

Convexifiability of continuous and discrete nonnegative quadratic programs for gap-free duality

“…inf(P 1 ) = max(D 1 ), between the problems (P 1 ) and (D 1 ) have been given using a generalized Karush-Kuhn-Tucker condition and copositivity of the related slack matrix [7,Theorem 5.1]. The semi-Lagrangian dual of nonconvex quadratic programs with quadratic and linear constraints is known to provide, in general, a better bound comparing to the standard Lagrangian dual and it admits a natural copositive program reformulation [7]. Copositive programs have been extensively studied in the framework of relaxation schemes for solving optimization problems in [1, 9, 11-13, 18, 26].…”