### Abstract

Original language | English |
---|---|

Pages (from-to) | 265-299 |

Journal | Mathematical Programming |

Volume | 149 |

Issue number | 1-2 |

Early online date | 16 Feb 2014 |

DOIs | |

Publication status | Published - Feb 2015 |

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### Keywords

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

### Cite this

*Mathematical Programming*,

*149*(1-2), 265-299. https://doi.org/10.1007/s10107-014-0750-8

}

*Mathematical Programming*, vol. 149, no. 1-2, pp. 265-299. https://doi.org/10.1007/s10107-014-0750-8

**Deriving robust counterparts of nonlinear uncertain inequalities.** / Ben-Tal, A.; den Hertog, D.; Vial, J.P.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Deriving robust counterparts of nonlinear uncertain inequalities

AU - Ben-Tal, A.

AU - den Hertog, D.

AU - Vial, J.P.

PY - 2015/2

Y1 - 2015/2

N2 - 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 - 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 - fenchal duality

KW - robust counterpart

KW - nonlinear inequality

KW - robust optimization

KW - support functions

U2 - 10.1007/s10107-014-0750-8

DO - 10.1007/s10107-014-0750-8

M3 - Article

VL - 149

SP - 265

EP - 299

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -