Deriving robust counterparts of nonlinear uncertain inequalities

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

Research output: Contribution to journalArticleScientificpeer-review

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
Pages (from-to)265-299
JournalMathematical Programming
Volume149
Issue number1-2
Early online date16 Feb 2014
DOIs
Publication statusPublished - Feb 2015

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Support Function
Uncertain Parameters
Fenchel Duality
Uncertainty
Conjugate functions
Convex Analysis
Nonlinear Constraints
Nonlinear Optimization
Convert
Nonlinear Problem
Duality
Optimization Problem
Calculate
Class

Keywords

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

Cite this

Ben-Tal, A. ; den Hertog, D. ; Vial, J.P. / Deriving robust counterparts of nonlinear uncertain inequalities. In: Mathematical Programming. 2015 ; Vol. 149, No. 1-2. pp. 265-299.
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Deriving robust counterparts of nonlinear uncertain inequalities. / Ben-Tal, A.; den Hertog, D.; Vial, J.P.

In: Mathematical Programming, Vol. 149, No. 1-2, 02.2015, p. 265-299.

Research output: Contribution to journalArticleScientificpeer-review

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AB - 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.

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