Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares

Behrends S, Schoebel A (2020)


Publication Type: Journal article

Publication year: 2020

Journal

Book Volume: 186

Pages Range: 911-935

Journal Issue: 3

DOI: 10.1007/s10957-020-01736-4

Abstract

Valid linear inequalities are substantial in linear and convex mixed-integer programming. This article deals with the computation of valid linear inequalities for nonlinear programs. Given a point in the feasible set, we consider the task of computing a tight valid inequality. We reformulate this geometrically as the problem of finding a hyperplane which minimizes the distance to the given point. A characterization of the existence of optimal solutions is given. If the constraints are given by polynomial functions, we show that it is possible to approximate the minimal distance by solving a hierarchy of sum of squares programs. Furthermore, using a result from real algebraic geometry, we show that the hierarchy converges if the relaxed feasible set is bounded. We have implemented our approach, showing that our ideas work in practice.

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How to cite

APA:

Behrends, S., & Schoebel, A. (2020). Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares. Journal of Optimization Theory and Applications, 186(3), 911-935. https://dx.doi.org/10.1007/s10957-020-01736-4

MLA:

Behrends, Soenke, and Anita Schoebel. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares." Journal of Optimization Theory and Applications 186.3 (2020): 911-935.

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