On convex relaxations for quadratically constrained quadratic programming

• Published in: Mathematical programming Vol. 136; no. 2; pp. 233 - 251
• Main Author: Anstreicher, Kurt M.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.12.2012; Springer; Springer Nature B.V
• Subjects: Applied sciences; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Envelopes; Estimates; Exact sciences and technology; Full Length Paper; Hulls; Hulls (structures); Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Quadratic forms; Quadratic programming; Studies; Theoretical; Semidefinite programming; 90C26; Convex envelope; Quadratically constrained quadratic programming; Reformulation-linearization technique; 90C22; Non convex programming; Constraint satisfaction; Semi definite programming; Quadratic programming; Convex programming; Convex hull; Diagonal matrix; Linearization techniques; Convex function; Objective function; Non convex analysis; Quadratic function; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-012-0602-3

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on is dominated by an alternative methodology based on convexifying the range of the quadratic form for . We next show that the use of “ BB” underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (“difference of convex”) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.