Dimitris bertsimas is the author of introduction to linear optimization 4. Robust queueing theory chaithanya bandi, dimitris bertsimasy, nataly youssefz we propose an alternative approach for studying queues based on robust optimization. It seeks to give a representative picture of the research topics most explored in recent years, highlight common themes in the investigations of independent research teams and highlight the contributions of rising as well as established researchers both to the theory of robust optimization and its practice. With robust lp, the feasible set is replaced by the robust feasible set. Robust design and optimization has even deeper roots in engineering.
In this work we try to bring the eld of robust optimization further, in an e ort to bring theory and practice one step closer together. Dimitris bertsimas author of introduction to linear. I thought that this one would have a more practical view to apply this method. A tutorial on robust optimization, given at the ima. Because of the wide and growing use of optimization in science, engineering, economics, and industry, it is essential for students and practitioners alike to develop an understanding of optimization algorithms. Motivated by this growing availability, we propose a novel schema for utilizing data to design uncertainty sets for robust optimization using statistical hypothesis tests. We provide an example showing that the feasibility set delimited by 2 is in general nonconvex, which contrasts with the convexity of robust linear programs.
Central themes in ro include understanding how to structure the uncertainty set r with loss probability p loss. This view of optimization as a process has become prominent. Robust linear optimization under general norms, with dessi pachamanova and melvyn sim, operations research letters, 32, 510516, 2004. We treat the case of both linear and nonlinear functions. Robust design optimization and design for six sigma, which is a quality improvement process leading to products conforming to six sigma quality. Adaptive distributionally robust optimization optimization online. Extensions of integer optimization includes chapters and 14, and treats mixed integer optimization and robust discrete optimization. While the methods of stochastic linear programming 1 may be regarded as a first approach to deal with uncertainties treating robustness as a side effect only, the notion of robust optimization gained focus in or after the publication of for an introductory paper, see also.
We model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability. We also refer the interested reader to the recent book of bental. Each of these decisions involves disjoint sets of variables and, possibly uncertain, coe cients. Introduction mathematical optimization leastsquares and linear programming convex optimization example course goals and topics nonlinear optimization brief history of convex optimization 11. Robust optimization is very useful for practice, since it is tailored to the information at hand, and it leads to computationally tractable formulations. Robust combinatorial optimization with variable budgeted. Unfortunately there is few books that deals with robust optimization. The key to our method is linking saa, distributionally robust optimization, and hypothesis testing of goodnessof. In these methods, the researcher has selected evolutionary algorithm, swarm intelligence and trajectory based method for optimizing the flexible manufacturing system layout using scheduling as. This book is devoted to robust optimization a specific and. Rivest clifford stein introduction to algorithms third edition the mit press cambridge, massachusetts london, england. Theory and applications of robust optimization 467 since there are more constraints to satisfy and the smaller the loss probability p loss.
In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a. Sim 2003, 2006 and convex constrained continuous minimax problems see zakovic. Bertsimas and sim 7 propose a different approach to control the level of conservatism in the solution that has the advantage that leads to a linear optimization. Both areas are practically significant as real world problems have very often both continuous and discrete variables and have elements of uncertainty that need to be addressed in a tractable manner. The so called minmaxmin robust optimization extends the classical minmax approach by. Optimization of linear functions with linear constraints is the topic of chapter 1, linear programming. Pdf sim, m robust discrete optimization and network flows. This chapter provides an introduction to optimization models and solution approaches. In section 2, we discuss a wide array of optimization classes and also uncertainty sets, and we consider the properties of the robust versions. Robust optimization a comprehensive survey sciencedirect. Strategy because robust design optimization simultaneously deals with optimization and robustness analysis, the computational effort becomes very high. We introduce a simple uncertainty set for arrival rates based on some structural properties of optimal queueing and routing solutions. In this work, we present a robust optimization method, which is suited for. A practicable framework for distributionally robust linear optimization dimitris bertsimas massachusetts institute of technology melvyn sim, meilin zhang nus business school, national university of singapore we developed a modular framework to obtain exact and approximate solutions to a class of linear optimiza.
Bertsimas and sim 34 propose an uncertainty set of the form. A robust optimization approach to inventory theory, with aurelie thiele, operations research, 54, 1, 150168, 2006. Distributionally robust optimization and its tractable approximations. Robust sample average approximation dimitris bertsimas. We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. The key philosophical reason behind the success of robust optimization has been the use of uncertainty.
A practicable framework for distributionally robust linear. Robust optimization for unconstrained simulationbased. The optimal point and optimal value of the new problem, however, may be quite di. A first course in optimization faculty server contact. Robust discrete optimization and network flows springerlink. Convex optimization and applications april 19, 2008. Dec 31, 20 the last decade witnessed an explosion in the availability of data for operations research applications. Maximizing or minimizing some function relative to some set, often representing a range of choices available in a certain situation. We also extend this improvement to a more general class of combinatorial optimization problems with interval uncertainty. In this thesis we introduce a robust optimization approach which is based on a binary minmaxmin problem. Theory and applications of robust optimization the university of. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Brown y, constantine caramanis z july 6, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the. The emphasis is on developing appropriate mathematical models to describe situations, implementing these models in a spreadsheet, using a spreadsheetbased solver.
Brown, and constantine caramanis overly conservative and in many cases are very similar to those produced by stochastic methods. This paper provides an overview of developments in robust optimization since 2007. Robust discrete optimization under ellipsoidal uncertainty sets, with melvyn sim, april 2004. Robust combinatorial optimization with variable budgeted uncertainty 3 protecting the associated ambiguous chance constraint with the same probability. The approach is flexible and widely applicable, and robust optimization problems built from our new sets are. It is therefore remarkable that reallife applications of robust optimization are still lagging behind. Optimization vocabulary your basic optimization problem consists of the objective function, fx, which is the output youre trying to maximize or minimize. It is necessary as well as bene cial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. Adaptive robust optimization for the security constrained unit commitment problem d bertsimas, e litvinov, xa sun, j zhao, t zheng ieee transactions on power systems 28 1, 5263, 2012. Bertsimas and sim 26 use this duality with a family of polyhedral sets that encode a budget of uncertainty. Byrne department of mathematical sciences university of massachusetts lowell a first course in optimization. However, as the resulting robust formulations involve conic quadratic problems see bental and nemirovski 4, such methods cannot be directly applied to discrete optimization.
1123 1533 601 735 1484 803 1487 305 1207 1111 156 1206 389 335 867 569 867 853 898 458 3 532 296 455 702 1204 1204 808 341 1047 438 438 696 747 981 996 937