SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs

• Main Authors: Nohra, Carlos J, Raghunathan, Arvind U, Sahinidis, Nikolaos V
• Format: Journal Article
• Language: English
• Published: 25-06-2021
• Subjects: Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.2106.13721

We consider the global optimization of nonconvex mixed-integer quadratic
programs with linear equality constraints. In particular, we present a new
class of convex quadratic relaxations which are derived via quadratic cuts. To
construct these quadratic cuts, we solve a separation problem involving a
linear matrix inequality with a special structure that allows the use of
specialized solution algorithms. Our quadratic cuts are nonconvex, but define a
convex feasible set when intersected with the equality constraints. We show
that our relaxations are an outer-approximation of a semi-infinite convex
program which under certain conditions is equivalent to a well-known
semidefinite program relaxation. The new relaxations are implemented in the
global optimization solver BARON, and tested by conducting numerical
experiments on a large collection of problems. Results demonstrate that, for
our test problems, these relaxations lead to a significant improvement in the
performance of BARON.