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We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main… (More)

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (" non-adjustable variables "), while the other part are variables that can be chosen after the realization (" adjustable variables "). We extend the Robust… (More)

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1-3]; El Ghaoui et al. [5,6]) to produce " robust " solutions of the… (More)

We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC).… (More)

In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on φ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems… (More)

Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefi-nite programming. For these cases, computationally… (More)

W e propose the use of robust optimization (RO) as a powerful methodology for multiperiod stochastic operations management problems. In particular, we study a two-echelon multiperiod supply chain problem, known as the retailer-supplier flexible commitment (RSFC) problem with uncertain demand that is only known to reside in some uncertainty set. We adopt a… (More)

We demonstrate that a conic quadratic problem min x e T x Ax ≥ b, A x − b 2 ≤ c T x − d , = 1, ..., m , y 2 = y T y, (CQP) is " polynomially reducible " to Linear Programming. We demonstrate this by constructing, for every ∈ (0, 1 2 ], an LP program (explicitly given in terms of and the data of (CQP)) min x,u e T x P x u + p ≥ 0 (LP) with the following… (More)

Dedicated to Jochem Zowe on the occasion of his 60th birthday. Abstract. We consider a conic-quadratic (and in particular a quadratically constrained) optimization problem with uncertain data, known only to reside in some uncertainty set U. The robust counterpart of such a problem leads usually to an NP-hard semidefinite problem; this is the case, for… (More)

We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions on this upper bounding convex function, a monotone convergence to a KKT… (More)