## Robust Semidefinite Programming” – in

Venue: | Handbook on Semidefinite Programming, Kluwer Academis Publishers |

Citations: | 38 - 18 self |

### BibTeX

@INPROCEEDINGS{Ben-tal_robustsemidefinite,

author = {Aharon Ben-tal and Laurent El and Ghaoui Arkadi Nemirovski},

title = {Robust Semidefinite Programming” – in},

booktitle = {Handbook on Semidefinite Programming, Kluwer Academis Publishers},

year = {},

pages = {139--162}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we consider semidefinite programs where the data is only known to belong to some uncertainty set U. Following recent work by the authors, we develop the notion of robust solution to such problems, which are required to satisfy the (uncertain) constraints whatever the value of the data in U. Even when the decision variable is fixed, checking robust feasibility is in general NP-hard. For a number of uncertainty sets U, we show how to compute robust solutions, based on a sufficient condition for robust feasibility, via SDP. We detail some cases when the sufficient condition is also necessary, such as linear programming or convex quadratic programming with ellipsoidal uncertainty. Finally, we provide examples, taken from interval computations and truss topology design. 1

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Citation Context ...egion U. The robust counterpart of the SDP is to minimize the worst-case value of the objective, among all robust solutions. This approach was introduced by the authors independently in [1, 2, 3] and =-=[15, 5]-=-; although apparently new in mathematical programming, the notion of robustness is quite classical in control theory (and practice). Even for simple uncertainty sets U, the resulting robust SDP is NP-... |

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Citation Context ...e admissible region U. The robust counterpart of the SDP is to minimize the worst-case value of the objective, among all robust solutions. This approach was introduced by the authors independently in =-=[1, 2, 3]-=- and [15, 5]; although apparently new in mathematical programming, the notion of robustness is quite classical in control theory (and practice). Even for simple uncertainty sets U, the resulting robus... |

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Citation Context ...of solutions, if any, to the equation Ax = b. Obtaining exact estimates for intervals of confidence for the elements of solutions x, even for the “linear interval algebra” problem, is already NP-hard =-=[17, 18]-=-. One classical approach to this problem resorts to interval calculus, where each one of the basic operations (+, −, x, /) is replaced by an “interval counterpart”, and standard (eg LU) linear algebra... |

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Citation Context ...e admissible region U. The robust counterpart of the SDP is to minimize the worst-case value of the objective, among all robust solutions. This approach was introduced by the authors independently in =-=[1, 2, 3]-=- and [15, 5]; although apparently new in mathematical programming, the notion of robustness is quite classical in control theory (and practice). Even for simple uncertainty sets U, the resulting robus... |

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