Comprehensive robust counterparts of uncertain problems a bental, s boyd, a nemirovski mathematical programming 107 12, 6389, 2006. Robust optimization ro is a relatively young methodology, developed mainly in the course of the last 15 years to analyze and optimize the performance of complex systems. Robust optimization leads to a tractable approach where an optimal static solution can be. Robust convexoptimization bental andnemirovski 1997, elghaoui et. Robust optimization is an important sub eld of optimization that deals with uncertainty in the data of optimization problems. Robust optimization is a common framework in optimization under uncertainty when the problem parameters are not known, but it is rather known that the parameters belong to some given uncertainty set. We make use of the multiobjective extension of rowwise uncertainty bental and nemirovski 1999, which is wellsuited to routing, and the polyhedral tra c model of benameur and kerivin 2005, which is commonly used in telecommunications. Robust optimization princeton series in applied mathematics series by aharon bental. Nonconvex robust optimization for problems with constraints, with omid nohadani and kwong meng teo, informs journal on computing preprint, 2009. A tractable approach for designing piecewise a ne policies. Pdf robust convex optimization aharon bental and arkadi.
Aharon bental is professor of operations research at the technion, israel institute for technology. 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. Bental and nemirovski approach to robust optimization consider the linear program min ct x p8 subject to ax. The paper surveys the main results of ro as applied to uncertain linear, conic quadratic and semidefinite programming. Robust optimization princeton series in applied mathematics series by aharon ben tal. Robust solutions of linear programming problems contaminated. By combining ideas from classical online algorithms developed in the computer science literature and robust and adaptive.
The reader is referred to bental and nemirovski 2008 and bertsimas et al. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the. Distributionally robust optimization and its tractable. Robust optimization by aharon bental overdrive rakuten. Sim nusdistributionally robust optimization26 aug 2009 4 47.
Contrast with classical robust optimization ro uncertainties in ro characterized by uncertainty set support bental and nemirovski 1998, bertsimas and sim 2004 j. Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. We then apply the robust optimization methodology bental and nemirovski. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and uptodate account of the subject. A soft robust model for optimization under ambiguity, with aharon ben tal and david b. One major motivation for studying robust optimization is that in many applications the data set is an appropriate notion of parameter uncertainty, e. The goal is to make a decision that is feasible no matter. The robust optimization method, which focused on treatability of computation in the case of data points disturbing in convex sets, was first proposed by soyster 2 and developed, respectively, by. A soft robust model for optimization under ambiguity 1222 operationsresearch584,part2of2,pp. This robust technique has obtained prodigious success since the late 1990s, especially in the. Robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems.
An essential book for anyone working on optimization and decision making under uncertainty, robust optimization also makes an ideal graduate textbook on the subject. For uncertain nonlinear optimization problems, the assumptions are also without. We also show how the notion of a budget of uncertainty enters into several di. With respect to portfolio selection, the major contributions. 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. Recent works using this general approach include bental and nemirovski1998,1999,2000,bertsimasandsim2004. The ensuing optimization problem is called robust optimization. The reader is referred to ben tal and nemirovski 2008 and bertsimas et al. Location allocation, and service frequency presented by maryam nikouei mehr, a candidate for the degree of master of science and hereby certify that, in their opinion, it is worthy of acceptance. Adaptive distributionally robust optimization dimitris bertsimas massachusetts institute of technology melvyn sim nus business school, national university of singapore. For these cases, computationally tractable robust counterparts of. The book is published by princeton university press, august 2009, see here. Authors usually adopt the robust convex optimization framework over an appropriate ambiguity set, and it is in this domain that our paper makes a contribution.
Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty. In many cases, a straightforward solution of the robust optimization problem of a certain type requires solving an optimization problem of a more complicated. In the robust optimization framework the problem solved is a minmax problem where a solution is judged according to its performance on the worst possible realization of the parameters. Robust optimization problems have first investigated by soyster 3 for linear programming problems and further studied by bental et al. Our focus will be on the computational attractiveness of ro approaches, as. A soft robust model for optimization under ambiguity, with aharon bental and david b. Arkadi nemirovski and publisher princeton university press. Contrast with classical robust optimization ro uncertainties in ro characterized by uncertainty set support ben tal and nemirovski 1998, bertsimas and sim 2004 j. In a way robust portfolio optimization brings ideas from taguchi robust engineering design to the design of portfolios. Robust optimization problems have first investigated by soyster 3 for linear programming problems and further studied by ben tal et al. Save up to 80% by choosing the etextbook option for isbn.
As defined by bental and nemirovski 1998, feasible solutions to 2 are robust feasible solutions and the optimal solution to 2 is a robust optimal solution. Robust optimization princeton series in applied mathematics 28 9780691143682. Robust optimization is still a relatively new approach to optimization problems affected by uncertainty. One major motivation for studying robust optimization is that in many applications the data set is an appropriate. Brown y, constantine caramanis z may 31, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the. Robust optimization 9780691143682, 9781400831050 vitalsource. Likelihood robust optimization for datadriven problems. By combining ideas from classical online algorithms developed in the. We propose a new tractable mixed integer linear formulation of the server problem that incorporates both information from the past and uncertainty about the future. 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. Bertsimas and sim 2004 introduced the concept price of robustness which they. Convex optimization, data uncertainty, robustness, linear programming, quadratic program. Under this framework, the objective and constraint functions are only assumed to belong to certain sets in function space the socalled \uncertainty sets.
Recent advances in robust optimization aharon bental minerva optimization center technion israel institute of technology robust optimization is a methodology for processing uncertain. Robust optimization is a young and active research. Aharon bental is professor of operations research at the. We demonstrate this phenomenon by studying 90 lps from the wellknown netlib collection. We demonstrate our approach for addressing a medical appointment scheduling problem as well as a multiperiod inventory control problem. This approach dates back to soyster 1973, who considered a deterministic linear optimization model that is feasible for all data lying in a convex set. Bental and nemirovski 1998, 1999, 2000, bertsimas and sim 2004, bertsimas and brown 2009, bertsimas et al. The essence of the problem is to make ordering, stocking, and. Recent advances in robust optimization researchgate. Optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed.
We revisit this example in more detail in section 4. We then apply the robust optimization methodology ben tal and nemirovski. Theory and applications of robust optimization dimitris bertsimas. Robust optimization methodology and applications springerlink. Aharon ben tal is professor of operations research at the technion, israel institute for technology.
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