Compact mixed-integer programming formulations in quadratic optimization

Benjamin Beach (), Robert Hildebrand () and Joey Huchette ()
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Benjamin Beach: Virginia Tech
Robert Hildebrand: Virginia Tech
Joey Huchette: Rice University

Journal of Global Optimization, 2022, vol. 84, issue 4, No 3, 869-912

Abstract: Abstract We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky (Neural Netw 94:103–114, 2017), formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. Further, we show that our formulation represents feasible points via a Gray code. We close with computational results on problems with quadratic objectives and/or constraints, showing that our proposed method (i) across the board outperforms existing MIP relaxations from the literature, and (ii) on hard instances produces better bounds than exact solvers within a fixed time budget.

Keywords: Quadratic optimization; Nonconvex optimization; Mixed-integer programming; Gray Code (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10898-022-01184-6

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