Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

CentER Discussion Paper Series No. 2012-053

32 Pages Posted: 3 Jul 2012

See all articles by Aharon Ben-Tal

Aharon Ben-Tal

Technion-Israel Institute of Technology

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

Jean-Philippe Vial

University of Geneva

Date Written: July 2, 2012

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

JEL Classification: C61

Suggested Citation

Ben-Tal, Aharon and den Hertog, Dick and Vial, Jean-Philippe, Deriving Robust Counterparts of Nonlinear Uncertain Inequalities (July 2, 2012). CentER Discussion Paper Series No. 2012-053. Available at SSRN: https://ssrn.com/abstract=2097765 or http://dx.doi.org/10.2139/ssrn.2097765

Aharon Ben-Tal (Contact Author)

Technion-Israel Institute of Technology ( email )

Technion City
Haifa 32000, Haifa 32000
Israel

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

Jean-Philippe Vial

University of Geneva ( email )

102 Bd Carl-Vogt
Genève, CH - 1205
Switzerland
(41) 22 705 81 24 (Phone)
(41) 22 705 81 04 (Fax)

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