## Robust Convex Quadratically Constrained Programs (2002)

Venue: | Mathematical Programming |

Citations: | 18 - 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 ...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 ...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 ...∈Rn } �Ax − b�2 , (45) where A = [a1, . . . , am] T ∈ R m×n and b ∈ R n . If m ≥ n and the matrix A has full column rank, the solution of this optimization problem is given by x ∗ = � A T A) −1 A T b =-=[13]-=-. Even when additional linear and convex quadratic constraints are imposed on the solution x, such as �x� 2 ≤ M, the linear least squares problem (45) is still a convex QCP. In many applications of le... |

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Citation Context ...− µ)(�µ − µ) T ] = (I − KC) T Σ(I − KC) + K T DK. (67) The non-robust version of this measurement model (i.e. Σ = 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 −... |

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Citation Context ...folio φ is given by rφ = r T φ. The objective is to choose a portfolio that maximizes some measure of “return” on the investment subject to appropriate constraints on the associated “risk”. Markowitz =-=[20, 21]-=- proposed a model for portfolio selection in which the “return” is the expected value E[rφ] of the portfolio return, the “risk” is the variance Var � � rφ of the return, and the optimal portfolio φ ∗ ... |

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Citation Context ...nvestigated a version of (5) in which the uncertainty structures Si are generalized ellipsoids and showed that the resulting robust optimization problem can be reduced to a semidefinite program (SDP) =-=[1, 22, 26]-=-. In this paper we explore uncertainty structures for which the corresponding robust problems can be reformulated as SOCPs. Our interest in this class of structures stems from the fact that both the w... |

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Citation Context ...nvestigated a version of (5) in which the uncertainty structures Si are generalized ellipsoids and showed that the resulting robust optimization problem can be reduced to a semidefinite program (SDP) =-=[1, 22, 26]-=-. In this paper we explore uncertainty structures for which the corresponding robust problems can be reformulated as SOCPs. Our interest in this class of structures stems from the fact that both the w... |

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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 ...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 ... themselves tractable optimization problems. Robustness as applied to uncertain least squares problems and uncertain semidefinite programs was independently studied by El Ghaoui and his collaborators =-=[11, 12]-=-. In keeping with the formulation proposed by Ben-Tal and Nemirovski, a generic robust convex QCP is given by minimize c T x subject to x T Qix + 2q T i x + γi ≤ 0, for all (Qi, qi, γi) ∈ Si, i = 1, .... |

139 | Second-order cone programming
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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 ... (x, ξ) ∈ K ⊂ R m , ∀ξ ∈ U, where ξ are the uncertain parameters, U is the uncertainty set, x ∈ R n is the decision vector, K is a convex cone and, for fixed ξ ∈ U, the function F (x, ξ) is K-concave =-=[4, 6]-=-. The optimization problem (4) is called a robust optimization problem. Ben-Tal and Nemirovski established that for certain classes of uncertainty sets U, robust counterparts of linear programs, quadr... |

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Citation Context ... themselves tractable optimization problems. Robustness as applied to uncertain least squares problems and uncertain semidefinite programs was independently studied by El Ghaoui and his collaborators =-=[11, 12]-=-. In keeping with the formulation proposed by Ben-Tal and Nemirovski, a generic robust convex QCP is given by minimize c T x subject to x T Qix + 2q T i x + γi ≤ 0, for all (Qi, qi, γi) ∈ Si, i = 1, .... |

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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 ...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 ... .. haM−1 . .. . . . . .. . .. . .. . .. . .. . .. . .. . .. 0 . . 0 hj0 hj1 hj2 hj3 . ⎤ ⎥ ⎥, ⎥ ⎦ ⎡ ⎢ gj = ⎢ ⎣ � 0 . . . . . . �� (n+m−1)×n 0 hj,m−1 � gj0 gj1 . gj,n−1 Under fairly general conditions =-=[24, 23]-=- the system of equations (58) has a solution provided p ≥ 2, i.e. on can find finite impulse response filters Gj(z), j = 1, . . . , p that can shorten the channel to any given target D(z). This equali... |

8 | Robust linear programming and optimal control
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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... |

7 |
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(Show Context)
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,... |

1 | Ghaoui. Minimum variance estimation with uncertain statistical model
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(Show Context)
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... |