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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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Journal ArticleDOI
TL;DR: This disclosure relates to a metering dispensing closure in which a pair of chambers are selectively placed in fluid communication with each other by an axially movable gravity actuated valve housed in one of the chambers.
Abstract: We consider the problem of dynamically cross-selling products (e.g., books) or services (e.g., travel reservations) in the e-commerce setting. In particular, we look at a company that faces a stream of stochastic customer arrivals and may offer each customer a choice between the requested product and a package containing the requested product as well as another product, what we call a “packaging complement.” Given consumer preferences and product inventories, we analyze two issues: (1) how to select packaging complements, and (2) how to price product packages to maximize profits. We formulate the cross-selling problem as a stochastic dynamic program blended with combinatorial optimization. We demonstrate the state-dependent and dynamic nature of the optimal package selection problem and derive the structural properties of the dynamic pricing problem. In particular, we focus on two practical business settings: with (the Emergency Replenishment Model) and without (the Lost-Sales Model) the possibility of inventory replenishment in the case of a product stockout. For the Emergency Replenishment Model, we establish that the problem is separable in the initial inventory of all products, and hence the dimensionality of the dynamic program can be significantly reduced. For both models, we suggest several packaging/pricing heuristics and test their effectiveness numerically.

97 citations

Journal Article
TL;DR: It is proved that under mild conditions an optimal solution is contained in a finite set and a basic scheme to enumerate this set is presented and improvements are suggested to reduce the number of function evaluations needed.
Abstract: In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Grobner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.

97 citations

Journal ArticleDOI
TL;DR: In this article, a bilinear variant of the Benders decomposition method was proposed to solve the chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance.
Abstract: In this paper, we study unit commitment (UC) problems considering the uncertainty of load and wind power generation. UC problem is formulated as a chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance. In addition to the conventional mixed integer linear programming formulation using Big-M, we present the bilinear mixed integer formulation of chance constraint, and then derive its linear counterpart using the McCormick linearization method. Then, we develop a bilinear variant of the Benders decomposition method, which is an easy-to-implement algorithm, to solve the resulting large-scale linear counterpart. Our results on typical IEEE systems demonstrate that (i) the bilinear mixed integer programming formulation is stronger than the conventional one and (ii) the proposed Benders decomposition algorithm is generally an order of magnitude faster than using a professional solver to directly compute both linear and bilinear chance-constrained UC models.

96 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions and prove convergence with probability one.
Abstract: In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Second, we use random models of higher quality than those produced by the usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability $\geq 1/2$. This is in contrast with stochastic gradient approaches, where the model is assumed to be “correct” only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Moreover, the new method is si...

96 citations

Journal ArticleDOI
TL;DR: In this article, a stochastic accelerated mirror-prox (SAMP) method was proposed for solving a class of monotone variational inequalities (SVI), which is based on a multi-step acceleration scheme.
Abstract: We propose a novel stochastic method, namely the stochastic accelerated mirror-prox (SAMP) method, for solving a class of monotone stochastic variational inequalities (SVI). The main idea of the proposed algorithm is to incorporate a multi-step acceleration scheme into the stochastic mirror-prox method. The developed SAMP method computes weak solutions with the optimal iteration complexity for SVIs. In particular, if the operator in SVI consists of the stochastic gradient of a smooth function, the iteration complexity of the SAMP method can be accelerated in terms of their dependence on the Lipschitz constant of the smooth function. For SVIs with bounded feasible sets, the bound of the iteration complexity of the SAMP method depends on the diameter of the feasible set. For unbounded SVIs, we adopt the modified gap function introduced by Monteiro and Svaiter for solving monotone inclusion, and show that the iteration complexity of the SAMP method depends on the distance from the initial point to the set of strong solutions. It is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems. We demonstrate the advantages of the SAMP method over some existing algorithms through our preliminary numerical experiments.

96 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023175
2022423
2021526
2020598
2019578
2018532