Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

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

Research output: Working paperDiscussion paperOther research output

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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

Fingerprint

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

Keywords

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

Cite this

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

Deriving Robust Counterparts of Nonlinear Uncertain Inequalities. / Ben-Tal, A.; den Hertog, D.; Vial, J.P.

Tilburg : Operations research, 2012. (CentER Discussion Paper; Vol. 2012-053).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

AU - Ben-Tal, A.

AU - den Hertog, D.

AU - Vial, J.P.

N1 - Pagination: 32

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Fenchel duality

KW - robust counterpart

KW - nonlinear inequality

KW - robust optimization

KW - support functions

M3 - Discussion paper

VL - 2012-053

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BT - Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

PB - Operations research

CY - Tilburg

ER -

Ben-Tal A, den Hertog D, Vial JP. Deriving Robust Counterparts of Nonlinear Uncertain Inequalities. Tilburg: Operations research. 2012. (CentER Discussion Paper).