Adaptive distributionally robust optimization dimitris bertsimas massachusetts institute of technology melvyn sim nus business school, national university of singapore. 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. 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. 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. Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty.
Robust optimization problems have first investigated by soyster 3 for linear programming problems and further studied by ben tal et al. Robust optimization is an important sub eld of optimization that deals with uncertainty in the data of optimization problems. Nonconvex robust optimization for problems with constraints, with omid nohadani and kwong meng teo, informs journal on computing preprint, 2009. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the.
This robust technique has obtained prodigious success since the late 1990s, especially in the. Robust optimization princeton series in applied mathematics series by aharon ben tal. In a way robust portfolio optimization brings ideas from taguchi robust engineering design to the design of portfolios. In many cases, a straightforward solution of the robust optimization problem of a certain type requires solving an optimization problem of a more complicated. 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. One major motivation for studying robust optimization is that in many applications the data set is an appropriate. The paper surveys the main results of ro as applied to uncertain linear, conic quadratic and semidefinite programming. 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. Theory and applications of robust optimization dimitris bertsimas. 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. Brown y, constantine caramanis z may 31, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the.
A tractable approach for designing piecewise a ne policies. A soft robust model for optimization under ambiguity, with aharon ben tal and david b. Robust optimization is still a relatively new approach to optimization problems affected by uncertainty. For uncertain nonlinear optimization problems, the assumptions are also without.
Pdf theory and applications of robust optimization researchgate. Our focus will be on the computational attractiveness of ro approaches, as. Distributionally robust optimization and its tractable. 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. The ensuing optimization problem is called robust optimization. Convex optimization, data uncertainty, robustness, linear programming, quadratic program. Robust optimization, robustness, adaptable optimization, applications of robust op. Likelihood robust optimization for datadriven problems. 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. Robust optimization leads to a tractable approach where an optimal static solution can be. Sim nusdistributionally robust optimization26 aug 2009 4 47. Bertsimas and sim 2004 introduced the concept price of robustness which they.
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. We demonstrate our approach for addressing a medical appointment scheduling problem as well as a multiperiod inventory control problem. Comprehensive robust counterparts of uncertain problems a bental, s boyd, a nemirovski mathematical programming 107 12, 6389, 2006. 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. Robust optimization by aharon bental overdrive rakuten. Robust optimization princeton series in applied mathematics 28 9780691143682. A soft robust model for optimization under ambiguity, with aharon bental and david b.
The essence of the problem is to make ordering, stocking, and. Aharon ben tal is professor of operations research at the technion, israel institute for technology. Aharon bental is professor of operations research at the technion, israel institute for technology. Robust optimization princeton series in applied mathematics series by aharon bental. Arkadi nemirovski and publisher princeton university press. Robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems. Recent works using this general approach include bental and nemirovski1998,1999,2000,bertsimasandsim2004. With respect to portfolio selection, the major contributions. 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 problems have first investigated by soyster 3 for linear programming problems and further studied by bental et al. The reader is referred to bental and nemirovski 2008 and bertsimas et al. Recent advances in robust optimization researchgate. The book is published by princeton university press, august 2009, see here. 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.
Under this framework, the objective and constraint functions are only assumed to belong to certain sets in function space the socalled \uncertainty sets. A soft robust model for optimization under ambiguity 1222 operationsresearch584,part2of2,pp. 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. The reader is referred to ben tal and nemirovski 2008 and bertsimas et al. Robust solutions of linear programming problems contaminated. 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. Robust optimization 9780691143682, 9781400831050 vitalsource. Bental and nemirovski 1998, 1999, 2000, bertsimas and sim 2004, bertsimas and brown 2009, bertsimas et al. An essential book for anyone working on optimization and decision making under uncertainty, robust optimization also makes an ideal graduate textbook on the subject. We demonstrate this phenomenon by studying 90 lps from the wellknown netlib collection.
Optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed. Aharon bental is professor of operations research at the. By combining ideas from classical online algorithms developed in the computer science literature and robust and adaptive. Contrast with classical robust optimization ro uncertainties in ro characterized by uncertainty set support ben tal and nemirovski 1998, bertsimas and sim 2004 j. 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. For these cases, computationally tractable robust counterparts of. Ben tal and nemirovski approach to robust optimization consider the linear program min ct x p8 subject to ax.
By combining ideas from classical online algorithms developed in the. Bental and nemirovski approach to robust optimization consider the linear program min ct x p8 subject to ax. The goal is to make a decision that is feasible no matter. Save up to 80% by choosing the etextbook option for isbn. One major motivation for studying robust optimization is that in many applications the data set is an appropriate notion of parameter uncertainty, e. Pdf robust convex optimization aharon bental and arkadi. 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. Robust optimization methodology and applications springerlink. Robust convexoptimization bental andnemirovski 1997, elghaoui et. Nemirovski we study convex optimization problems for which the data is not speci ed 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. We also show how the notion of a budget of uncertainty enters into several di. We revisit this example in more detail in section 4.
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