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

Carlos J. Nohra; Arvind U. Raghunathan; Nikolaos V. Sahinidis

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arXiv · 2106.13721

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

Carlos J. Nohra·Arvind U. Raghunathan·Nikolaos V. Sahinidis

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Abstract

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.

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Carlos J. Nohra, Arvind U. Raghunathan, Nikolaos V. Sahinidis. 2021-06-25. SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs. https://arxiv.org/abs/2106.13721

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