Abstract
We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush–Kuhn–Tucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are exact for an extended CDT (two-ball trust-region) problem. Importantly, the sufficient conditions can be verified by solving linear optimization problems.
Highlights
Consider the following nonconvex quadratic optimization problem, which is referred to as the extended trust region problem: (P)min x∈Rn x Q0x + 2q0 x subject to x Q1x + 2q1 x ≤ 1 Ax − a 2 ≤ 1Bx ≤ b, where Q0, Q1 are (n × n) symmetric matrices, A is an ( × n) matrix, B is an (m × n) matrix, a ∈ R, b ∈ Rm and q0, q1 ∈ Rn
It has been shown that exactness of Lagrangian relaxation can fail for the CDT problem, or for the trust region problem with only one additional linear inequality constraint
Equivalent copositive reformulations for many important problems are known, among them quadratic optimization problems under a mild assumption [2,3,13], and some special optimization problems under uncertainty [4,18,32,37]. It has been shown in [7] that, for quadratic optimization problems with additional nonnegative constraints, copositive relaxations provides a tighter bound than the usual Lagrangian relaxation
Summary
Consider the following nonconvex quadratic optimization problem, which is referred to as the extended trust region problem:. It has been shown that exactness of Lagrangian (or SDP) relaxation can fail for the CDT problem, or for the trust region problem with only one additional linear inequality constraint. 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] 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.
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