Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

A. Ben-Tal, D. den Hertog, J.P. Vial

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Abstract

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.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages32
Volume2012-053
Publication statusPublished - 2012

Publication series

NameCentER Discussion Paper
Volume2012-053

Keywords

  • Fenchel duality
  • robust counterpart
  • nonlinear inequality
  • robust optimization
  • support functions

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    Ben-Tal, A., den Hertog, D., & Vial, J. P. (2012). Deriving Robust Counterparts of Nonlinear Uncertain Inequalities. (CentER Discussion Paper; Vol. 2012-053). Operations research.