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

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-KuhnTucker 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.

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

“…Now we see that (x l ) i ∈ {0, 1} for all l ∈ K + and for all i ∈ B. To see this, from the last relation of (9),…”

“…It is widely known that convexity of sets and functions of optimization problems underpins many important developments of mathematical theory and methods of optimization. For instance, recent research (see [9,23,24]) has examined the role of convexity in duality and exact conic programming relaxations for special classes of quadratic programs such as extended trust-region problems, CDT problems (two-balls trust-region problems) and separable minimax quadratic programs. When it comes to studying duality for hard nonconvex quadratic programs, such as general nonnegative quadratic programs with quadratic constraints and mixed integer quadratic programs, identifying the key features that underline the zero duality gap property and then finding classes of quadratic programs that possess the features and zero duality gaps are undoubtedly important.…”

Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong lagrangian duality

“…Then, the algorithm finds a global optimal solution as the KKT point with the smallest objective value. A geometric condition ensuring exact copositive relaxation for problem (P) with indefinite B, positive semidefinite C and additional linear inequality constraints is given in [12]. In [7,32], the authors derived a second order cone programming (SOCP) relaxation for problem (P) where the quadratic forms are simultaneously diagonalizable.…”

A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem

“…Then Bomze et al [7] proved some new sufficient and necessary conditions for both local and global optimality and gave a complete characterization in the degenerate case. Besides, they provided some verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are equivalent for an extended (CDT) problem [8] . Bienstock [6] showed how to adapt a construction of Barvinok's method [2] so as to obtain a polynomial-time algorithm for the quadratic program with a fixed number of quadratic constraints.…”