Stochastic programming second edition peter kall institute for operations research and mathematical methods of economics university of zurich ch8044 zurich stein w. Easy to perform for linear programming uses duality theory but, this is a postmortem tool. Stochastic versus robust optimization for a transportation problem. Stochastic programming the science that provides us with tools to design and control stochastic systems with the aid of mathematical programming techniques lies at the intersection of statistics and mathematical programming. We also refer the interested reader to the recent book of bental. Re24 cently the worst case approach attracted considerable attention and became known 25 as distributionally robust stochastic optimization drso. The problem is to determine the number of vehicles to book at. While applications of stochastic programming have been reported over many years in the technical literature, and a number of applications of robust optimization have appeared recently, nearly all of these have been laborintensive individual projects. T1 a robust optimization perspective on stochastic programming. They cover three approaches to convex optimization with uncertain input data. Comparison of approximations in stochastic and robust optimization.
In linear optimization, bertsimas and sim 2004 and. Stochastic programming and robust optimization are optimization tools deal. A basic difficulty of solving such stochastic optimization problems is that the i. Provides applications from the fields of robust optimal control design in case of stochastic uncertainty. Robust optimization was relatively recently introduced as a method to incorporate uncertainty into mathematical programming models bental et al. A robust optimization perspective of stochastic programming, operations research, 344357556, 10581071. Uncertain parameters are often represented by scenarios in. The simplest example is markowitz portfolio optimization. The robust optimization framework developed by lin et al. This book is devoted to robust optimization a specific and relatively.
Distributionally robust stochastic programming siam. In this approach, u is a userprovided convex compact uncertainty set and the constraint is to hold for all z. A robust optimization perspective on stochastic programming xin chen. Robust and stochastic optimization 1 course description formulation and treatment of in nite and nitedimensional robust and stochastic programs. What are the main differences between stochastic optimization. In this paper we study distributionally robust stochastic programming in a setting 8 where there is a speci ed reference probability measure and the. Distributionally robust stochastic programming 3 84 otherwise. Hows robust optimization and stochastic programming used.
Bental and nemirovski 2000 obtain probability bounds against constraint violation by assuming. These deviation measures capture distributional asymmetry and lead to. Assume that is well defined and finite valued for all this implies that for every. An attractive feature of the framework is that we convert the original model into a secondorder cone program, which is computationally tractable both in theory and in. In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty.
Pdf theory and applications of robust optimization researchgate. Hybrid robust and stochastic optimization for closedloop. The robust convex programming problem is convex but it is numerically hard to solve because of in nite number of constraints. Robust portfolio optimization and management frontmatter page i wednesday, july 25, 2007 1. Comparison of approximations in stochastic and robust. Dec 15, 2017 unit commitment seeks the most cost effective generator commitment schedule for an electric power system to meet net load, defined as the difference between the load and the output of renewable generation, while satisfying the operational constraints on transmission system and generation resources. A tractable approximation of stochastic programming via robust optimization xin chen.
The book stochastic programming is a comprehensive introduction to the field and its basic mathematical tools. In contrast to ro, stochastic optimization starts by assuming the uncertainty has a probabilistic. In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations. Optimization problems arising in practice involve random model parameters. Scenario generation using latin hypercube sampling with backward reduction. Robust optimization with ambiguous stochastic constraints. Normally when you think of optimization, here are following things you can follow to improve the formulation of the problem. Information about possible values of the problem data is incorporated in the model, and the model generates solutions that are less sensitive to data uncertainty. Because of our goal to solve problems of the form 1. Distributionally robust stochastic programming siam journal. The key idea is to hedge the solutions against worstcase realizations of the uncertain parameters. For models that truly require the most general methods, it uses simulation optimization, again using psi technology for vectorized evaluation of simulation trials at.
Postek et al robust optimization with ambiguous stochastic constraints under mean and dispersion information 2 them is robust optimization. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy. This text presents the main complexity theorems in convex optimization and their algorithms. Features optimization problems that in practice involve random model parameters. Stochastic programming and robust optimization mie utoronto. For models with the required structure, it can apply robust optimization and stochastic programming methods to solve the problem. Boeing professor of operations research, sloan school of management and operations research center, massachusetts. Such models deal, in a constructive manner, with noisy, incomplete or uncertain data. May 16, 2006 abstract in this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations. For the computation of robust optimal solutions, i. Section 2 gives an introduction on optimization under uncertainty, and presents brief comparisons among the wellknown subfields of optimization under uncertainty such as ro, stochastic programming sp, and fuzzy optimization fo. It provides an introduction to stochastic programming and robust optimization models.
However, heavycomputationalburden has prevented wider applicationsin practice. The robust optimization methods propose some relaxation techniques to deal with such a. Robust optimization methods are much more recent, with most research in the past ten years. A uni ed framework for stochastic optimization warren b. Stochastic programming and robust optimization are the most widely studied approaches for unit. Our major contribution is to develop a novel hybrid robuststochastic programming hrsp approach to simultaneously model two different types of uncertainties by including stochastic scenarios for transportation. May 14, 2019 as far as i know, there is no difference. Boeing professor of operations research, sloan school. These are partial notes from ece 6990, robust and stochastic optimization, as taught by prof. Hybrid robust and stochastic optimization for closedloop supply chain network design using accelerated benders decomposition. Stochastic optimization methods applications in engineering. Robust optimization an overview sciencedirect topics. Stochastic programming and robust optimization springerlink. The proposed robust formulations have the advantage to be solvable in polynomial time and to have theoretical guarantees for the quality of their solutions, which is not the case for the stochastic formulation.
Protocols, performance, and control,jagannathan sarangapani 26. Here is a nonempty closed subset of, is a random vector whose probability distribution is supported on a set. A marriage of robust optimization and stochastic programming melvyn sim decision sciences, nus business school. A robust optimization model for stochastic logistic problems. In the following, we present in brief the general robust optimization approach, which explicitly takes into account the various forms of. Robust stochastic approximation approach to stochastic. A novel hybrid robuststochastic programming approach. Aproximations of stochastic and robust optimization programs. Aproximations of stochastic and robust optimization programs 3 throughout, we assume that there exists a feasible solution to rcp. Ee364b convex optimization ii stanford engineering everywhere. Help the students to understand how to model uncertainty into mathematical optimization problems, what uncertainty brings to the decision process and which techniques help to manage uncertainty in solving the problems. Pdf implementing and comparing stochastic and robust. Kuhn, daniel, wolfram wiesemann, and angelos georghiou.
Stochastic programming is a framework for modeling optimization problems that involve uncertainty. Pdf traditional optimisation tools focus on deterministic problems. Robust optimization is a subset of stochastic optimization. Robust optimization and stochastic programming approaches. N2 in this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations. Robust and stochastic optimization cornell university. Theory and applications of robust optimization citeseerx. Eilyan bitarat cornell university in the fall of 2015. In the article, they showed that the solution to any robust optimization problem is also a solution to a drsp.
The remainder of the chapter is organized as follows. Stochasticversus robustoptimization fora transportationproblem. Under an assumed probability distribution of the uncertain parameters, the objective function becomes a collection of random variables. Decentralized convex optimization via primal and dual decomposition. Convex approximation techniques and performance bounds. Box 2110 n6402 molde, norway reference to this text is peter kall and stein w. Primal and dual linear decision rules in stochastic and robust optimization. With an introduction to stochastic control theory, second edition,frank l. In this paper we consider optimization problems where the objective function is given in a form of the expectation. Optimization under uncertainty 291 stochastic programming 293 dynamic programming 308 robust optimization 312 summary 332 chapter 11. Includes numerous references to stochastic optimization, stochastic programming and its applications to engineering, operations research and economics. Stochastic optimization starts by assuming the uncertainty has a probabilistic description. We have stochastic and deterministic linear programming, deterministic and stochastic network. This paper considers robust optimization ro, a more recent approach to optimization under uncertainty, in which the uncertainty model is not stochastic, but rather deterministic and setbased.
Applications to multistage robust and stochastic optimization will also be considered. Accelerated benders decomposition with valid inequalities and paretooptimal cuts. In stochastic programming it goes back at least to z a ckov a 25. Using a linear decision rule, we also propose a tractable approximation approach for solving a class of multistage chanceconstrained stochastic linear optimization problems. In the simplest theoretical formulation, you know the mean returns and covariance matrix of all the securities in your universe, so you can just do the optimization and get the right answe. Hows robust optimization and stochastic programming used in. A tractable approximation of stochastic programming via. A linear decision based approximation approach to stochastic programming, operations research, 562, 344357. Operations research 432 1995a 264281 and mulvey and ruszczynski operations research 43 3 1995b 477490.
Stochastic linear programming and robust optimization models are. Although this book mostly covers stochastic linear programming since that is the best developed topic, we also discuss stochastic nonlinear programming, integer programming and network. A robust optimization perspective to stochastic models. Robust optimization, stochastic programming, and simulation. In the framework of twostage stochastic programming, is given by the optimal value of the corresponding secondstage problem. In this paper we study distributionally robust stochastic programming in a setting where there is a specified reference probability measure and the uncertainty set of probability measures consists of measures in some sense close to the reference measure.
More details can be found in classical textbooks both. Stochastic optimization starts by assuming the uncertainty has a probabilistic. Starting from the fundamental theory of blackbox optimization, the material progresses towards recent advances in structural and stochastic optimization. Selected applications in areas such as control, circuit design. Stochastic programming, handbooks in or and ms, elsevier, ruszczynski and shapiro, 2003. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. Stochastic optimization methods kurt marti springer. The topic of uncertainty in optimization was identified already by george dantzig, the father of linear programming and an icon of operations. Abstract in this chapter, the authors give a brief introduction to important concepts of ro paradigm. The uncertainty can be explicitly taken into account through preventive approaches such as twostage stochastic programming, parametric programming, fuzzy programming, chance constraint programming, robust optimization techniques, and conditional value at risk verderame et al. A robust optimization perspective on stochastic programming. Pdf robust stochastic approximation approach to stochastic. The first one to introduce the concept of adjustable robustness was ben tal et al.
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