## Robust Solutions To Uncertain Semidefinite Programs (1998)

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Venue: | SIAM J. OPTIMIZATION |

Citations: | 77 - 8 self |

### BibTeX

@ARTICLE{Ghaoui98robustsolutions,

author = {Laurent El Ghaoui and Francois Oustry and Hervé Lebret},

title = {Robust Solutions To Uncertain Semidefinite Programs},

journal = {SIAM J. OPTIMIZATION},

year = {1998},

volume = {9},

number = {1},

pages = {33--52}

}

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### Abstract

In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

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Citation Context ... converges to the minimum norm solution of the nominal problem (24). Remark. In this case, the RSDP is a regularized version of the nominal SDP, which belongs to the class of Tikhonov regularizations =-=[34]. Th-=-e regularization parameters44 L. EL GHAOUI, F. OUSTRY, AND H. LEBRET is 2ρ and is chosen according to some a priori information on uncertainty associated with the nominal problem’s data. Taking ρ ... |

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Citation Context ...R m −{0} and the symmetric matrices Fi = F T i ∈ Rn×n ,i =0,...,m, are given. SDPs are convex optimization problems and can be solved in polynomial time with, e.g., primal-dual interior-point met=-=hods [24, 35, 26, 19, 2]. SD-=-Ps include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., [12, 1, 35, 22]. In the SDP (1), the “data” co... |

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Citation Context ...ual interior-point methods [24, 35, 26, 19, 2]. SDPs include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., =-=[12, 1, 35, 22]. In-=- the SDP (1), the “data” consist of the objective vector c and the matrices F0,...,Fm. In practice, these data are subject to uncertainty. An extensive body of work has concentrated on the sensiti... |

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Citation Context ..., and “tractable counterparts” of a large class of uncertain SDPs are given. The case of robust linear programming, under quite general assumptions on the perturbation bounds, is studied in detail=-= in [6]-=-. Our paper can be seen as a complement of [8], giving ways to cope with (not necessarily) tractable robust SDPs by means of upper bounds. (In particular, our paper handles the case of nonlinear depen... |

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Citation Context ...utions via SDP. Links between regularity of solutions and robustness are, of course, expected. One of our side objectives is to clarify these links to some extent. This paper extends results given in =-=[16]-=- for the least-squares problem. The approach developed here can be viewed as a special case of stochastic programming in which the distribution of the perturbation is uniform. The ideas developed in t... |

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Citation Context ...ual interior-point methods [24, 35, 26, 19, 2]. SDPs include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., =-=[12, 1, 35, 22]. In-=- the SDP (1), the “data” consist of the objective vector c and the matrices F0,...,Fm. In practice, these data are subject to uncertainty. An extensive body of work has concentrated on the sensiti... |

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Citation Context ...rturbation problem include [15, 9, 29]. We are not aware of special references for general SDPs with randomly perturbed data except for the last section of [30], some exercises in the course notes of =-=[13]-=-, and section 2.6 in [23]. The main objective of this paper is to quantify the effect of unknown but bounded deterministic perturbation of problem data on solutions. In our framework, the perturbation... |

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Citation Context ...on problems by Shapiro [31] and Bonnans, Cominetti, and Shapiro [10]. Shortly after completion of our manuscript, we became aware of the ongoing work of Ben-Tal and Nemirovski on the same subject. In =-=[7]-=-, they apply similar ideas to a truss topology design problem and derive very efficient algorithms for solving the corresponding robustness problem. In [8], the general problem of tractability of obta... |

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Citation Context ...R m −{0} and the symmetric matrices Fi = F T i ∈ Rn×n ,i =0,...,m, are given. SDPs are convex optimization problems and can be solved in polynomial time with, e.g., primal-dual interior-point met=-=hods [24, 35, 26, 19, 2]. SD-=-Ps include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., [12, 1, 35, 22]. In the SDP (1), the “data” co... |

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Citation Context ...ues and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include [30, 31, 10] for general nonlinear programs, [33] for semi-infinite programs, and =-=[32]-=- for semidefinite programs. When the perturbation affecting the data of the problem is not necessarily small, a sensitivity analysis is not sufficient. For general optimization problems, a whole field... |

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Citation Context ...esimal, and regularity of optimal values and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include [30, 31, 10] for general nonlinear programs, =-=[33]-=- for semi-infinite programs, and [32] for semidefinite programs. When the perturbation affecting the data of the problem is not necessarily small, a sensitivity analysis is not sufficient. For general... |

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Citation Context ...table robust SDPs by means of upper bounds. (In particular, our paper handles the case of nonlinear dependence of the data on the uncertainties.) A unified treatment, and more results, will appear in =-=[4]-=-. The paper is divided as follows. Our problem is defined in section 2. In section 3, we show how to compute upper bounds on our problem via SDP. We give special attention to the so-called full pertur... |

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Citation Context ...R m −{0} and the symmetric matrices Fi = F T i ∈ Rn×n ,i =0,...,m, are given. SDPs are convex optimization problems and can be solved in polynomial time with, e.g., primal-dual interior-point met=-=hods [24, 35, 26, 19, 2]. SD-=-Ps include linear programs and convex quadratically constrained quadratic programs, and arise in a wide range of engineering applications; see, e.g., [12, 1, 35, 22]. In the SDP (1), the “data” co... |

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Citation Context ...sarily unique and/or regular) solutions of the nominal problem. Problem (25) is particularly suitable to the recent so-called U-Newton algorithms for solving problem (24). These methods, described in =-=[21, 25], req-=-uire that the Hessian of the “smooth part” (the so-called U-Hessian) of the objective of (24) be positive definite. For general mappings F(·), this property is not guaranteed. However, when looki... |

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Citation Context ... in general, but sometimes can be easily tested in practical examples, as seen in section 5. We note that H3(a) implies that R(x) �= 0 for every x. Hypothesis H1 is equivalent to Robinson’s condit=-=ion [27], which can be expressed-=- in terms of � � T T F(x) − τLL R(x) F(x, τ)= . R(x) τI Robinson’s condition is stated in [27] as the existence of x0 ∈ R m , τ0 ∈ R such that � , 0 ∈ int � F(x0,τ0)+dF(x0,τ0)R... |

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Citation Context ...h the perturbations are assumed to be infinitesimal, and regularity of optimal values and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include =-=[30, 31, 10]-=- for general nonlinear programs, [33] for semi-infinite programs, and [32] for semidefinite programs. When the perturbation affecting the data of the problem is not necessarily small, a sensitivity an... |

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Citation Context ...h the perturbations are assumed to be infinitesimal, and regularity of optimal values and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include =-=[30, 31, 10]-=- for general nonlinear programs, [33] for semi-infinite programs, and [32] for semidefinite programs. When the perturbation affecting the data of the problem is not necessarily small, a sensitivity an... |

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Citation Context ...ming in which the distribution of the perturbation is uniform. The ideas developed in this paper draw mainly from two sources: control theory, in which we have found the tools for robustness analysis =-=[36, 17, 12]-=- and some recent work on sensitivity analysis of optimization problems by Shapiro [31] and Bonnans, Cominetti, and Shapiro [10]. Shortly after completion of our manuscript, we became aware of the ongo... |

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Citation Context ... impact of, say, a random objective on the distribution of optimal values (this problem is called the “distribution problem”). References relevant to this approach to the perturbation problem incl=-=ude [15, 9, 29]-=-. We are not aware of special references for general SDPs with randomly perturbed data except for the last section of [30], some exercises in the course notes of [13], and section 2.6 in [23]. The mai... |

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Citation Context ... impact of, say, a random objective on the distribution of optimal values (this problem is called the “distribution problem”). References relevant to this approach to the perturbation problem incl=-=ude [15, 9, 29]-=-. We are not aware of special references for general SDPs with randomly perturbed data except for the last section of [30], some exercises in the course notes of [13], and section 2.6 in [23]. The mai... |

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Citation Context ...h the perturbations are assumed to be infinitesimal, and regularity of optimal values and solution(s), as functions of the data matrices, is studied. Recent references on sensitivity analysis include =-=[30, 31, 10]-=- for general nonlinear programs, [33] for semi-infinite programs, and [32] for semidefinite programs. When the perturbation affecting the data of the problem is not necessarily small, a sensitivity an... |

8 |
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Citation Context ...plicit SDPs that yield robust counterparts to SOCPs nonconservatively are given for a wide class of uncertainty structures. In some cases, albeit not all, the robust counterpart is itself an SOCP. In =-=[16, 14], -=-the special case of least-squares problems with uncertainty in the data is studied at length. 5.6. Robust maximum norm minimization. Several engineering problems take the form (27) where minimize �H... |

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Citation Context ...e [15, 9, 29]. We are not aware of special references for general SDPs with randomly perturbed data except for the last section of [30], some exercises in the course notes of [13], and section 2.6 in =-=[23]-=-. The main objective of this paper is to quantify the effect of unknown but bounded deterministic perturbation of problem data on solutions. In our framework, the perturbation is not necessarily small... |

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Citation Context ...e given p×q matrices. A well-known instance of this problem is the linear least-squares problem, with H(x)=Ax − b. Another example is a minimal norm extension problem for a Hankel operator studied =-=in [18], -=-in which H0 is a given (arbitrary) n × n Hankel matrix and Hi, i =1,...,m is the n × n Hankel matrix associated with the polynomial 1/z i . In practice, the matrices Hi, i =0,...,m are subject to pe... |

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Citation Context ...ogram is readily written as an SDP by introducing slack variables. In fact, the robust LP is a second-order cone program (SOCP) for which efficient specialpurpose interior-point methods are available =-=[24, 20, 23]-=-. We note that hypothesis H3 holds blockwise. This yields the following result. Theorem 5.2. The optimal value of the robust LP can be computed by solving the convex problem (23). If the latter satisf... |

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Citation Context ...� ≤e T i x + fi, i=1,...,L, where Ci ∈ R ni×m , di ∈ R ni , ei ∈ R m , fi ∈ R, i =1,...,L. SOCPs can be formulated as SDPs, but special-purpose, more efficient algorithms can be devised f=-=or them; see [24, 5, 23]. As-=-suming that Ci,di,ei,fi are subject to linear—or even rational—uncertainty, we may formulate the corresponding RSDP as an SDP. This SDP can be written as an SOCP if the uncertainty is unstructured... |

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Citation Context ...ult of Shapiro [31, Thm. 3.1], by which we conclude that the solution is actually Lipschitz stable (inequality (18) holds with the exponent 1/2 replaced by 1). Finally, using the results from Attouch =-=[3], w-=-e can show that computing the solution for ρ → 0 amounts to selecting the minimum norm solution among the solutions of the nominal problem. Theorem 5.3. The optimal value of the min-max problem (24... |

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Citation Context ... impact of, say, a random objective on the distribution of optimal values (this problem is called the “distribution problem”). References relevant to this approach to the perturbation problem incl=-=ude [15, 9, 29]-=-. We are not aware of special references for general SDPs with randomly perturbed data except for the last section of [30], some exercises in the course notes of [13], and section 2.6 in [23]. The mai... |