The optimal point and optimal value of the new problem, however, may be quite di. Tilburg university adjustable robust optimization zhen, jianzhe. Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty. Finished well, finished the parts i intended to read a while ago so i dont remember much else. 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. Aug 10, 2009 robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems. Robust discrete optimization under ellipsoidal uncertainty sets, with melvyn sim, april 2004.
The basic idea is to seek a solution which is immunized against the effect of data uncertainty. In that section, we also describe extensions of our work to more general uncertainty sets, e. Theory and applications of robust optimization dimitris bertsimas. Convex optimization and applications april 19, 2008. Robust optimization is an emerging area in research that allows addressing different optimization problems and specifically industrial optimization problems where there is a degree of uncertainty in some of the variables involved. An attractive aspect of their method is that the new robust formulation is also a linear optimization problem. Submitted to the sloan school of management on may 14, 2004, in partial fulfillment of the requirements for the degree of. Brown y, constantine caramanis z july 6, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the. Theory and applications of robust optimization 2011 siam rev. Robust combinatorial optimization with variable budgeted uncertainty 3 protecting the associated ambiguous chance constraint with the same probability.
Table 1 presents the tractable robust counterparts of an uncertain linear optimization problem for different classes of uncertainty sets. Our focus will be on the computational attractiveness of ro approaches, as well. A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. The second motivation of robust optimization is to model stochastic and dynamic optimization problems using uncertainty sets. Nemirovski princeton university press, 2009 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 this relatively new approach to. Sim, \ robust discrete optimization and network ows, mathematical programming, b98. Robust optimization made easy with rome harvard business. 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.
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. Introduction to interval and robust optimization uncertainty in electricity markets and system operation july 4, 2016. All the algorithms and methods have been coded in a matlab toolbox allowing numerical analysis, reliability analysis, simulation, sensitivity, optimization, robust design and much more. These robust counterparts are derived using the three steps that are described above. 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 noesis solutions noesis solutions.
Robust optimisation math bibliographies in harvard style. Under this framework, the objective and constraint functions are only assumed to belong to certain sets in function space the socalled \uncertainty sets. Dec 31, 20 the last decade witnessed an explosion in the availability of data for operations research applications. 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 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. Distributionally robust optimization and its tractable approximations. Boyd at stanford, this is a rare case where watching live lectures is better than reading a book. Robust linear optimization under general norms, with dessi pachamanova and melvyn sim, operations research letters, 32, 510516, 2004. Robust optimization made easy with rome 974 operations research 594, pp.
We also show that the robust counterpart of an nphard. Rome runs in the matlab environment, so that users can take full advantage of the numerical and graphical capabilites of matlab for preprocessing and analysis of data. Robust discrete optimization and network flows springerlink. However, as the resulting robust formulations involve conic quadratic problems see bental and nemirovski 4, such methods cannot be directly applied to discrete optimization. Robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems. Contrast with classical robust optimization ro uncertainties in ro characterized by uncertainty set support bental and nemirovski 1998, bertsimas and sim 2004 j.
Robust transportation network design under user equilibrium. We model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability. In this paper, we propose a framework for robust modeling of linear programming problems using uncertainty sets described by an arbitrary norm. Jan 12, 2015 robust optimization is a young and active research field that has been mainly developed in the last 15 years. Central themes in ro include understanding how to structure the uncertainty set r with loss probability p loss. The foundation of our paper can be found in bertsimas and sim 2004, which introduce the notion ofabudgetofuncertaintytoreducetheconservatism of robust optimization.
Milton stewart school of industrial and systems engineering georgia institute of technology, atlanta georgia 303320205 usa november 2012. We improve the wellknown result presented in bertsimas and sim math program b98. Specifically, robust second order cone problems socps remain scops and robust semidefinite optimization problems sdps remain sdps. Their excellent teaching and discussion inspired my interest in the field of optimization. Robust sample average approximation dimitris bertsimas. Robust optimization with multiple ranges optimization online. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. Optimization of linear functions with linear constraints is the topic of chapter 1, linear programming.
Robust queueing theory chaithanya bandi, dimitris bertsimasy, nataly youssefz we propose an alternative approach for studying queues based on robust optimization. However, we need conic duality instead of lp duality in step 2 to derive the tractable robust counterparts for the conic uncertainty set. This book is devoted to robust optimization a specific and. Strategy because robust design optimization simultaneously deals with optimization and robustness analysis, the computational effort becomes very high. The goal is to make a decision that is feasible no matter. Byrne department of mathematical sciences university of massachusetts lowell a first course in optimization. Robust design and optimization mathematical optimization. Brown, and constantine caramanis overly conservative and in many cases are very similar to those produced by stochastic methods. The optimization of nonlinear functions begins in chapter 2 with a more complete treatment of maximization of unconstrained functions that is covered in calculus. Adaptive distributionally robust optimization management. Abstract robust optimization is an effective method to tackle optimization problems under data uncertainty.
A practical guide to robust optimization sciencedirect. In this paper we survey the primary research, both theoretical and applied, in the area of robust optimization ro. I engineering applications, which presents some new applications of different methods, and ii applications in various areas, where recent contributions. We explicitly characterize the robust counterpart as a convex optimization problem that involves the dual norm of the given norm. His final research area involves bringing the fields of operations researchparticularly optimization and statisticstogether. The key contribution is a tractable theory of optimization under uncertainty. While robust optimization has been applied in the references above as a way to address parameter uncertainty, bertsimas and thiele 9 use this framework to 2. This optimal solution is insensitive to small design changes so that if small perturbations of the design parameters occur, the performance of the system. In a similar spirit, bertsimas and sim 34 propose an uncertainty set of the form. Sim, m robust discrete optimization and network flows. It is therefore remarkable that reallife applications of robust optimization are still lagging behind.
There are several ways to apply robust optimization and the choice of form is typical of the problem that is being solved. Practical robust optimization an introduction lnmb. This book covers stateoftheart optimization methods and their applications in wide range especially for researchers and practitioners who wish to improve their knowledge in this field. Christian artigues laas laboratory for analysis and architect. Robust design optimization and design for six sigma, which is a quality improvement process leading to products conforming to six sigma quality. Released under the lgpl license, the engine can be used, modified and redistributed free of charge.
Hence, in contrast to a stochastic optimization model, a robust optimization model does not directly account for risk preferences. Robust optimization is an important sub eld of optimization that deals with uncertainty in the data of optimization problems. Tongxin, adaptive robust optimization for the security constrained unit commitment problem, ieee transactions on power systems, vol. He has been teaching a doctoral class entitled robust optimization for the last several years based on his work in the area. Robust optimisation math bibliographies cite this for me. 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. 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. Ever since 2000, the robust optimization area has witnessed a burst of research activity in both theory and application. One major motivation for studying robust optimization is that in many applications the data set is an appropriate notion of parameter uncertainty, e. With robust lp, the feasible set is replaced by the robust feasible set. Both risk and ambiguity should be taken into account in modeling an optimization problem under uncertainty. Robust optimization by melvyn sim submitted to the sloan school of management on may 14, 2004, in partial ful. To obtain a tractable formulation, we approximate the adaptive distributionally robust optimization problem using linear decision rule ldr techniques.
Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty. Introduction to linear optimization athena scientific. Theory and applications of robust optimization 467 since there are more constraints to satisfy and the smaller the loss probability p loss. Recently, bertsimas and sim proposed a different approach for robust linear optimization with polyhedral as opposed to ellipsoidal uncertainty sets. Despite these recent advances in robust optimization the ory and techniques, there has been a conspicuous lack of. Dimitris bertsimas is the boeing leaders for global operations professor of management, a professor of operations research, and the associate dean for the master of business analytics at mit. Rome robust optimization made easy is an algebraic modeling toolbox designed to solve a class of robust optimization problems. This approach was conceptualized by bertsimas and sim 16 for continuous problems. Sim, \robust discrete optimization and network ows, mathematical programming, b98. Dimitris bertsimas is the author of introduction to linear optimization 4.
Whereas stochastic programming assumes there is a probabilistic description of the uncertainty, robust optimization works with a deterministic, setbased description of the uncertainty. Lncs 3064 a robust optimization approach to supply chain. Our focus is on the computational attractiveness of ro approaches, as well as the modeling power and broad applicability of the methodology. Wireless network design all the presented results are strongly based on discussions with experts from our industrial partners, such as. Robust optimization princeton series in applied mathematics 9780691143682. Robust linear optimization under general norms sciencedirect. Theory and applications of robust optimization siam. A faculty member since 1988, his research interests include optimization, stochastic systems, machine learning, and their application. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. There are several ways to apply robust optimization and the choice. We utilize the approach in 5,6, which leads to linear robust. Robust optimization is a relatively new approach to modeling uncertainty in optimization problems. Bertsimas and sim show that the robust counterpart is still an lp. Robust optimization is very useful for practice, since it is tailored to the information at hand, and it leads to computationally tractable formulations. We also refer the interested reader to the recent book of bental. Sim nusdistributionally robust optimization26 aug 2009 4 47.
Bertsimas likes his equations though and is sometimes a bit short on explanations. Professor dimitris bertsimas massachusetts institute of. The approach is flexible and widely applicable, and robust optimization problems built from our new sets are. While there are many highoptimization ro came from the robust control community, ro is a distinct.
A practicable framework for distributionally robust linear. We also extend this improvement to a more general class of combinatorial optimization problems with interval uncertainty. 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. A robust optimization approach to inventory theory, with aurelie thiele, operations research, 54, 1, 150168, 2006. What are some good resources to learn about optimization.
I also want to thank professor melvyn sim, professor chungpiaw teo, professor jie sun and professor karthik natarajan for their mentorship and friendship. Introduction to linear optimization by dimitris bertsimas. We show that the adaptive distributionally robust linear optimization problem can be formulated as a classical robust optimization problem. Thresholded covering algorithms for robust and maxmin. In such an optimization scheme, a measure for the robustness of the system or component is used as optimization constraint or objective in order to meet the best robust performance possible.
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. Dimitris bertsimas author of introduction to linear. The robust optimization approach constructs a solution that is feasible for any realization of. Robust combinatorial optimization with variable budgeted. Optimization algorithms methods and applications intechopen. A tutorial on robust optimization, given at the ima.
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