Topic
Stochastic programming
About: Stochastic programming is a research topic. Over the lifetime, 12343 publications have been published within this topic receiving 421049 citations.
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TL;DR: In this paper, a retailer decides its level of involvement in the futures market and in the pool as well as the selling price offered to its potential clients with the goal of maximizing the expected profit at a given risk level.
Abstract: This paper presents a bilevel programming approach to solve the medium-term decision-making problem faced by a power retailer. A retailer decides its level of involvement in the futures market and in the pool as well as the selling price offered to its potential clients with the goal of maximizing the expected profit at a given risk level. Uncertainty on future pool prices, client demands, and rival-retailer prices is accounted for via stochastic programming. Unlike in previous approaches, client response to retail price and competition among rival retailers are both explicitly considered in the proposed bilevel model. The resulting nonlinear bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear programming problem by replacing the lower-level optimization by its Karush-Kuhn-Tucker optimality conditions and converting a number of nonlinearities to linear equivalents using some well-known integer algebra results. A realistic case study is solved to illustrate the efficient performance of the proposed methodology.
233 citations
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TL;DR: The key feature of the proposed approach is that the on-line optimization problem is solved off-line via parametric programming techniques, and only simple function evaluations are required, to obtain the optimal inputs to the plant for the current state of the plant.
233 citations
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TL;DR: It is proved that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program, and a general projection property for multivariate distributions with given means and covariances is proved.
Abstract: We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. We first prove a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance robust objective. This allows us to use known univariate results in the multidimensional setting, and to add new results in this direction. In particular, we characterize a general class of objective functions (the so-called one- or two-point support functions), for which the robust objective is reduced to a deterministic optimization problem in one variable. Finally, we adapt a result from Geoffrion (1967a) to reduce the main problem to a parametric quadratic program. In particular, our results are true for increasing concave utilities with convex or concave-convex derivatives. Closed-form solutions are obtained for special discontinuous criteria, motivated by bonus- and commission-based incentive schemes for portfolio management. We also investigate a multiproduct pricing application, which motivates extensions of our results for the case of nonnegative and decision-dependent returns.
232 citations
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TL;DR: This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem and results compare the performance of the algorithm to MINOS 5.0.
Abstract: This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.
232 citations
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TL;DR: This work presents a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPPs) and demonstrates an optimal rate of convergence not only in terms of its dependence on the number of the iteration, but also on a variety of problem parameters.
Abstract: We present a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPPs). The basic idea of this algorithm is to incorporate a multistep acceleration scheme into the primal-dual method without smoothing the objective function. For deterministic SPP, the APD method achieves the same optimal rate of convergence as Nesterov's smoothing technique. Our stochastic APD method exhibits an optimal rate of convergence for stochastic SPP not only in terms of its dependence on the number of the iteration, but also on a variety of problem parameters. To the best of our knowledge, this is the first time that such an optimal algorithm has been developed for stochastic SPP in the literature. Furthermore, for both deterministic and stochastic SPP, the developed APD algorithms can deal with the situation when the feasible region is unbounded, as long as a saddle point exists. In the unbounded case, we incorporate the modified termination criterion introduced b...
232 citations