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The Flexible Γ-Approach for Nonlinear Discrete and Nonlinear Combinatorial Optimization

Dennis Adelhütte(dennis.adelhuette***at***fau.de)
Jana Dienstbier(jana.jd.dienstbier***at***fau.de)
Frauke Liers(frauke.liers***at***fau.de)

Abstract: The flexible Gamma-approach has been introduced for adjusting the degree of conservatism in robust counterparts. It has mainly been applied to linear combinatorial optimization problems: Instead of aiming for solutions which are optimal regardless of how the uncertainties manifest, the objective is to ensure robustness against Gamma uncertainties. The contribution of this paper is a generalization of this approach for (mixed-integer) nonlinear optimization problems. We study the cases in which the functions considered are concave or linear, as well as non-concave, in the uncertainty. By applying reformulation techniques that have been established for nonlinear inequalities under uncertainty, we derive equivalent robust counterparts. In both cases the computational tractability of the counterpart depends on the structure of the geometry of the uncertainty set. We explicitly present robust counterparts for combinatorial problems, e.g. the Quadratic Assignment Problem. We conduct computational studies for the Gamma-robust Quadratic Assignment Problem and the Gamma-robust Vehicle Routing Problem with Soft Time Windows to demonstrate the computational tractability in terms of solution quality and running time.

Keywords: Discrete Optimization, Flexible Gamma-Approach, Combinatorial Optimization, Mixed-Integer Nonlinear Optimization, Robust Optimization, Gamma-Uncertainty

Category 1: Robust Optimization

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Entry Submitted: 05/25/2020
Entry Accepted: 05/25/2020
Entry Last Modified: 05/25/2020

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