Milton stewart school of industrial and systems engineering at the georgia institute of technology. In the main part of the paper we show that if u is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems linear. Hunter academic chair, school of industrial and systems engineering, georgia. Selected topics in robust convex optimization springerlink. The uncertainty set u can be replaced by its convex hull convu, i. The robust counterpart for a linear program lp with polyhedral. Robust and reliable portfolio optimization formulation of. In a robust optimization approach, the uncertain parameters are assumed to belong to some uncertainty set. A tight characterization of the performance of static. Faculty of industrial engineering and management, technionisrael institute of technology, technion city, haifa 32000, israel. Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty. The ensuing optimization problem is called robust optimization.
He serves on the editorial boards of several journals, including mathematics of operations research, siam journal on optimization, journal of convex analysis, and mathematical modeling and numerical algorithms. Robust optimisation math bibliographies cite this for me. Robust convex optimization mathematics of operations research. Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself andor its solution. Pdf we study convex optimization problems for which the data is not. Aharon bental arkadi nemirovski robust optimization methodology and applications received. Cuttingset methods for robust convex optimization with pessimizing. For these cases, computationally tractable robust counterparts of. Idea behind robust optimization is to consider the worst cas e scenario without a speci c distribution assumption. Theory and applications of robust optimization dimitris bertsimas. Robust convex optimization mathematics of operations.
Proceedings of 20 th ifip tc7 conference on system modelling and optimization, july 2327, trier, germany, 2001b. In this paper, we overview several selected topics in this popular area, specifically, 1 recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, 2 tractability of robust counterparts, 3 links between ro and traditional chance constrained settings of problems with stochastic data, and 4. September 12, 2001 published online february 14, 2002 springerverlag 2002 abstract. Bental and nemirovski approach to robust optimization consider the linear program min ct x p8 subject to ax. Aharon bental is a professor at the technionisrael institute of technology and head of the minerva optimization center. The paper surveys the main results of ro as applied to uncertain linear, conic quadratic and semidefinite programming. Robust truss topology optimization under uncertain loads. 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. Arkadi nemirovski born march 14, 1947 is a professor at the h. He has been a leader in continuous optimization and is best known for his work on the ellipsoid method, modern interiorpoint methods and robust optimization. A tutorial on robust optimization, given at the ima, march 11, 2003.
Online firstorder framework for robust convex optimization. Analysis, algorithms, and engineering applications mpssiam series on optimization aharon bental, arkadi nemirovski lectures on convex optimization is devoted to well structured and efficiently solvable convex optimization problems, with an emphasis on conic quadratic and semidefinite programming. Robust optimization ro isa modeling methodology, combined with computational tools, to. In the following, bental and nemirovskis results on costs are generalized to smooth convex conic problems under lipschitz uncertainty, given reasonably mild regularity conditions. Robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems.
Analysis, algorithms, and engineering applications conn, andrew r. Convex optimization, data uncertainty, robustness, linear programming. Zhen et al adjustable robust optimization via fouriermotzkin elimination article submitted. Our results parallel and extend the work of elghaoui and lebret on robust least squares, and the work of bental and nemirovski on robust conic convex optimization problems. Ever since 2000, the robust optimization area has witnessed a burst of research activity in both theory and application. Arkadi nemirovski also is a professor at the technionisrael. We now list a few of the complexity results that are relevant in the sequel. The theory presented here is useful for desensitizing solutions to illconditioned problems, or for computing solutions that guarantee a certain performance in the. In this paper we lay the foundation of robust convex optimization. As shown in bental and nemirovski 2000, in many cases the price of robustness. Mathematics of operations research 23 4, 769805, 1998. This is because in order to solve harc, one needs to find an optimal decision rule with respect to both.
The basic idea is to seek a solution which is immunized against the effect of data uncertainty. Robust convex optimization article pdf available in mathematics of operations research 234 november 1998 with 718 reads how we measure reads. However, as the resulting robust formulations involve conic quadratic. Adjustable robust optimization via fouriermotzkin elimination. T opology design ttd problem for more details, see bental and nemirovski 1997. Theory and applications of robust optimization the university of. Nonconvex robust optimization for problems with constraints. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. Bental and nemirovski 14 introduced a number of impor.
Robust convex optimization 1998 mathematics of operations research. For linear optimization, bental and nemirovski 15 propose a robust model. Ifip tc7 20th conference on system modeling and optimization july 2327, 2001, trier, germany springer us aharon bental, arkadi nemirovski auth. Abstract robust optimization is an effective method to tackle optimization problems under data uncertainty. Tilburg university adjustable robust optimization zhen, jianzhe. Nemirovski we study convex optimization problems for which the data is not speci. Selected topics in robust convex optimization optimization online. We consider a general worstcase robust convex optimization problem, with arbitrary. In a robust optimization approach, the uncertain parame. When are static and adjustable robust optimization. Topology design ttd problem for more details, see bental and nemirovski 1997. Robust linear optimization under general norms mit. Optimization for machine learning university of texas at. Cuttingset methods for robust convex optimization with.
Comprehensive robust counterparts of uncertain problems a bental, s boyd, a nemirovski mathematical programming 107 12, 6389, 2006. We study convex optimization problems for which the data is not specified exactly and it. Robust optimization is a young and active research field that has been mainly. The roots of robust optimization can be found in the eld of ro bust control and in the work of soyster 9 as well as later works by bental and nemirovski 1,2 and independently by elghaoui and lebret 6 and elg haoui et al 7. Approximate optimality conditions for robust convex. This paper considers robust optimization ro, a more recent. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Robust solutions to l1, l2, and linfinity uncertain. Robust convexoptimization bental andnemirovski 1997, elghaoui et. X corollary 1 can be used to reduce the complexity of solving adjustable robust optimization problems. For these cases, computationally tractable robust counterparts. Bental, aharon and nemirovski, arkadi, lectures on modern convex optimization. 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.
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