Compact mixed-integer programming formulations in quadratic optimization

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 mixe...

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Bibliographic Details
Published in:Journal of global optimization Vol. 84; no. 4; pp. 869 - 912
Main Authors: Beach, Benjamin, Hildebrand, Robert, Huchette, Joey
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2022
Springer
Springer Nature B.V
Subjects:
Approximation
Computer Science
Continuity (mathematics)
Integer programming
Linear programming
Mathematics
Mathematics and Statistics
Mixed integer
Operations Research/Decision Theory
Optimization
Perturbation methods
Quadratic equations
Real Functions
Tightness
Gray Code
Nonconvex optimization
Quadratic optimization
Mixed-integer programming
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary: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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-022-01184-6