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Dual approach for two-stage robust nonlinear optimization

Frans J.C.T. de Ruiter (fjctderuiter***at***gmail.com)
Jianzhe Zhen (jianzhe.zhen***at***epfl.ch)
Dick den Hertog (d.denHertog***at***uvt.nl)

Abstract: Adjustable robust minimization problems in which the adjustable variables appear in a convex way are difficult to solve. For example, if we substitute linear decision rules for the adjustable variables, then the model becomes convex in the uncertain parameters, whereas for computational tractability we need concavity in the uncertain parameters, which implies that the adjustable variables should appear linearly in the objective and constraints. In this paper we reformulate the original adjustable robust nonlinear problem with a polyhedral uncertainty set into an equivalent adjustable robust linear problem, for which all existing approaches for adjustable robust linear problems can be used. The reformulation is obtained by first dualizing over the adjustable variables and then over the uncertain parameters. The polyhedral structure of the uncertainty set then appears in the linear constraints of the dualized problem, and the nonlinear functions of the adjustable variables in the original problem appear in the uncertainty set of the dualized problem. We show how to recover the linear decision rule to the original primal problem and extend our method to multistage cases. This paper also describes how to effectively obtain lower bounds (for minimization problems) on the optimal objective value by linking the realizations in the uncertainty set of dualized problem to realizations in the original uncertainty set. Two numerical examples, one on a distribution problem on a network with commitments, and one on finding the equilibrium of a system with piecewise-linear springs, show the effectiveness and applicability of the new approach.

Keywords: Adjustable robust optimization, nonlinear inequalities, duality, linear decision rules.

Category 1: Robust Optimization

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Entry Submitted: 03/12/2018
Entry Accepted: 03/12/2018
Entry Last Modified: 11/27/2019

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