## Robust Convex Quadratically Constrained Programs (2002)

Venue: | Mathematical Programming |

Citations: | 21 - 2 self |

### BibTeX

@ARTICLE{Goldfarb02robustconvex,

author = {D. Goldfarb and G. Iyengar},

title = {Robust Convex Quadratically Constrained Programs},

journal = {Mathematical Programming},

year = {2002},

volume = {97},

pages = {495--515}

}

### OpenURL

### Abstract

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.

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Citation Context ...rb † G. Iyengar ‡ July 1st, 2002 Abstract In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski =-=[4]-=-. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We ... |

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Citation Context ... j� , (47) where without any loss of generality �v� ≤ 1, �u� ≤ 1, and ξ j ∼ N (0, Ω j ), j = 1, . . . , l. Without the stochastic term, the uncertainty set (47) has the affine structure considered in =-=[4, 5, 6]-=-. The term � l j=1 uj ξ j models the imperfect knowledge of the stochastic perturbations in a – the decision maker knows the total variance and modes Ω j but does not know the variance of each of the ... |

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Citation Context ... � c0 bT 0 � p� − � ci bT i � � 0. b0 A0 i=1 τi bi Ai Moreover, if p = 1 then the converse holds if there exists x0 such that F1(x0) > 0. For a discussion of the S-procedure and its applications, see =-=[7]-=-. Since v = 0 is strictly feasible for 1 − v T Gv ≥ 0, the S-procedure implies that (28) holds for all 1 − vT Gv ≥ 0 if and only if there exists a τ ≥ 0 such that � M = ν − τ − yT 0 (F0 + ηN)y0 −ryT 0... |

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Citation Context ...that the matrix Q is positive semidefinite. Suppose Q � 0. Then Q = V T V for some V ∈ R m×n and the quadratic constraint x T Qx ≤ −(2qT x + γ) is equivalent to the second-order cone (SOC) constraint =-=[2, 18, 22]-=- �� � � 2Vx � � (1 + γ + 2qT �� ���� ≤ 1 − γ − 2q x) T x. (2) ∗ Submitted to Math Programming, Series B. Please do not circulate. † IEOR Department, Columbia University, Email: gold@ieor.columbia.edu.... |

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Citation Context ...that the matrix Q is positive semidefinite. Suppose Q � 0. Then Q = V T V for some V ∈ R m×n and the quadratic constraint x T Qx ≤ −(2qT x + γ) is equivalent to the second-order cone (SOC) constraint =-=[2, 18, 22]-=- �� � � 2Vx � � (1 + γ + 2qT �� ���� ≤ 1 − γ − 2q x) T x. (2) ∗ Submitted to Math Programming, Series B. Please do not circulate. † IEOR Department, Columbia University, Email: gold@ieor.columbia.edu.... |

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Citation Context ...Pattern classification using hyperplanes is called linear discrimination. In a typical application of linear discrimination, the hyperplane (w, b) is chosen by solving the following quadratic program =-=[8, 19, 28]-=-. minimize 1 2 �w�2 + C �� i=1 ξi � , subject to wT xi + b ≥ 1 − ξi, if yi = +1, w T xi + b ≥ 1 + ξi, if yi = −1, ξi ≥ 0, i = 1, . . . , l. Instead of solving (40), one typically solves its dual given... |

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Citation Context ...ainty sets that meet this criterion. Adding robustness reduces the sensitivity of the optimal decision to fluctuations in the parameters and can often result in significant improvement in performance =-=[3, 16, 27]-=-. Typically, the complexity of the deterministic reformulation of the robust problem is higher than the non-robust version of the problem. However, since the worst case complexity of convex QCPs is co... |

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Citation Context ...oblems are typically sensitive to parameter fluctuations, the errors in the input parameters tend to get amplified in the decision vector. This, in turn, leads to a sharp deterioration in performance =-=[3, 10]-=-. The problem of choosing a decision vector in the presence of parameter perturbation was formalized by Ben-Tal and Nemirovski [4, 5] as follows: minimize c T x subject to F (x, ξ) ∈ K ⊂ R m , ∀ξ ∈ U,... |

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Citation Context ...= Σ0 and D = D0 for fixed Σ0 and D0) is the well-known Gaussian linear stochastic model [14]. The robust measurement model developed here is a variant of the model proposed by Calafiore and El Ghaoui =-=[9]-=- where the a priori covariance Σ was known exactly and the noise covariance D ∈ � D : D −1 = D −1 0 + L∆R + RT ∆ T L T � 0, �∆� ≤ 1 � . They show that the problem of choosing the gain matrix K to mini... |