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Deriving robust counterparts of nonlinear uncertain inequalities

Aharon Ben-Tal(abental***at***ie.technion.ac.il)
Dick den Hertog(D.denHertog***at***uvt.nl)
Jean-Philippe Vial(jphvial***at***gmail.com)

Abstract: In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

Keywords: Fenchel duality, robust counterpart, nonlinear inequality, robust optimization, support functions

Category 1: Robust Optimization

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Linear, Cone and Semidefinite Programming

Citation: CentER Discussion Paper 2012-0053 Tilburg University Tilburg, The Netherlands July, 2012

Download: [PDF]

Entry Submitted: 07/03/2012
Entry Accepted: 07/03/2012
Entry Last Modified: 07/03/2012

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