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sample average approximation method FOR A CLASS OF STOCHASTIC VARIATIONAL INEQUALITY PROBLEMS

Journal of Systems Science & Complexity 2011年第6期24卷 1143-1153页

作者： Mingzheng WANG Guihua LIN Yuli GAO M. Montaz ALI Institute of Systems Engineering Dalian University of Technology Dalian 116024 China School of Mathematical Sciences Dalian University of Technology Dalian 116024 China School of Computational and Applied Mathematics University of the Witwatersrand Wits-2050 Johannesburg South Africa

This paper considers a class of stochastic variational inequality problems. As proposed by Jiang and Xu （2008）, by using the so-called regularized gap function, the authors formulate the problems as constrained optimization problems and then propose a sample average approximation method for solving the problems. Under some moderate conditions, the authors investigate the limiting behavior of the optimal values and the optimal solutions of the approximation problems. Finally, some numerical results are reported to show efficiency of the proposed method.

关键词： Convergence gap function sample average approximation method stochastic variational inequality.

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sample average approximation method for a class of stochastic generalized Nash equilibrium problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 261卷 387-393页

作者： Li, Pei-Yu Southwest Petr Univ Sch Sci Chengdu 610500 Peoples R China

We consider a class of stochastic generalized Nash equilibrium problems (SGNEPs). Under some mild conditions, we reformulate the SGNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce an interior point method to solve the SAA problems. Comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic generalized Nash equilibrium model in river basin pollution game. (C) 2013 Published by Elsevier B.V.

关键词： Stochastic generalized Nash equilibrium problem sample average approximation method Interior point method

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A smooth penalty-based sample average approximation method for stochastic complementarity problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 287卷 20-31页

作者： He, Suxiang Wei, Min Tong, Hengqing Wuhan Univ Technol Sch Sci Wuhan 430070 Peoples R China

sample average approximation method is one of the effective methods in the stochastic optimization. A smooth penalty-based sample average approximation method for stochastic nonlinear complementarity problems is presented in this paper. Based on a smooth penalty function, a reformulation is proposed for the equivalent problem of EV formulation for stochastic complementary problems and it is proven that its solutions are existent under some mild assumptions. An implementable sample average approximation method for the reformulation is further established and its convergence is analyzed. The numerical results for some test examples are reported at last to show efficiency of the proposed method. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Stochastic complementarity problems sample average approximation method Penalty function Stationary point Convergence

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A Nonlinear Lagrange Algorithm for Stochastic Minimax Problems Based on sample average approximation method

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JOURNAL OF APPLIED MATHEMATICS 2014年第unknown期000卷 1-8页

作者： He, Suxiang Nie, Yunyun Wang, Xiaopeng Wuhan Univ Technol Sch Sci Wuhan 430070 Peoples R China

An implementable nonlinear Lagrange algorithm for stochastic minimax problems is presented based on sample average approximation method in this paper, in which the second step minimizes a nonlinear Lagrange function with sample average approximation functions of original functions and the sample average approximation of the Lagrange multiplier is adopted. Under a set of mild assumptions, it is proven that the sequences of solution and multiplier obtained by the proposed algorithm converge to the Kuhn-Tucker pair of the original problem with probability one as the sample size increases. At last, the numerical experiments for five test examples are performed and the numerical results indicate that the algorithm is promising.

关键词： NONLINEAR theories LAGRANGE equations STOCHASTIC analysis PROBLEM solving CHEBYSHEV approximation sample average approximation method ALGORITHMS

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Validation analysis of mirror descent stochastic approximation method

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MATHEMATICAL PROGRAMMING 2012年第2期134卷 425-458页

作者： Lan, Guanghui Nemirovski, Arkadi Shapiro, Alexander Univ Florida Gainesville FL 32611 USA Georgia Inst Technol Atlanta GA 30332 USA

The main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation (SA) type algorithms. To this end we show that while running a Mirror Descent Stochastic approximation procedure one can compute, with a small additional effort, lower and upper statistical bounds for the optimal objective value. We demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method, while their computational cost is considerably smaller.

关键词： Stochastic approximation sample average approximation method Stochastic programming Monte Carlo sampling Mirror descent algorithm Prox-mapping Optimality bounds Large deviations estimates Asset allocation problem Conditional value-at-risk

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Sampling average approximation method for a class of stochastic Nash equilibrium problems

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OPTIMIZATION methodS & SOFTWARE 2013年第4期28卷 785-795页

作者： Li, Pei-Yu He, Zhi-Feng Lin, Gui-Hua Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Shanghai Univ Sch Management Shanghai 200444 Peoples R China

We consider a class of stochastic Nash equilibrium problems (SNEP). Under some mild conditions, we reformulate the SNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce a semismooth Newton method to solve the SAA problems. The comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas-oil markets.

关键词： stochastic Nash equilibrium problem sample average approximation method semismooth Newton method

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Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case

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JOURNAL OF GLOBAL OPTIMIZATION 2015年第3期62卷 481-505页

作者： Bonnel, Henri Collonge, Julien Univ New Caledonia ERIM Noumea 98851 New Caledonia

Our paper consists of two main parts. In the first one, we deal with the deterministic problem of minimizing a real valued function over the Pareto outcome set associated with a deterministic convex bi-objective optimization problem (BOP), in the particular case where depends on the objectives of (BOP), i.e. we optimize over the Pareto set in the outcome space. In general, the optimal value of such a kind of problem cannot be computed directly, so we propose a deterministic outcome space algorithm whose principle is to give at every step a range (lower bound, upper bound) that contains . Then we show that for any given error bound, the algorithm terminates in a finite number of steps. In the second part of our paper, in order to handle also the stochastic case, we consider the situation where the two objectives of (BOP) are given by expectations of random functions, and we deal with the stochastic problem of minimizing a real valued function over the Pareto outcome set associated with this Stochastic bi-objective Optimization Problem (SBOP). Because of the presence of random functions, the Pareto set associated with this type of problem cannot be explicitly given, and thus it is not possible to compute the optimal value of problem . That is why we consider a sequence of sample average approximation problems (SAA-, where is the sample size) whose optimal values converge almost surely to as the sample size goes to infinity. Assuming nondecreasing, we show that the convergence rate is exponential, and we propose a confidence interval for . Finally, some computational results are given to illustrate the paper.

关键词： Optimization over the Pareto image set Multi-objective deterministic optimization Multi-objective stochastic optimization Multi-objective convex optimization sample average approximation method

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Fourth-party logistics network design with service time constraint under stochastic demand

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JOURNAL OF INTELLIGENT MANUFACTURING 2023年第3期34卷 1203-1227页

作者： Yin, Mingqiang Huang, Min Qian, Xiaohu Wang, Dazhi Wang, Xingwei Lee, Loo Hay Northeastern Univ Coll Informat Sci & Engn State Key Lab Synthet Automat Proc Ind Shenyang 110819 Liaoning Peoples R China Shenzhen Univ Coll Management Res Inst Business Analyt & Supply Chain Managemen Shenzhen 518060 Peoples R China Northeastern Univ Coll Comp Sci & Engn Shenyang 110819 Liaoning Peoples R China Natl Univ Singapore Dept Ind & Syst Engn 10 Kent Ridge Crescent Singapore 119260 Singapore

Due to the increasingly competitive nature of the global market, the capability of controlling delivery time is becoming a significant advantage for enterprises. A novel fourth-party logistics (4PL) network design problem with the objective of minimizing the overall cost under service time constraint and stochastic demand is proposed in the paper. To address this problem, a two-stage nonlinear stochastic programming model is proposed. The topological structure of the 4PL network is decided in the first stage, while the network flows are determined in the second stage. By using auxiliary variables to linearize the service time constraint and by adopting the sample average approximation (SAA) method to handle the stochastic demand, the two-stage nonlinear stochastic programming model is transformed into a mixed integer linear programming (MILP) model. To overcome the difficulties of solving the MILP model caused by a large number of demand scenarios and integer-valued decision variables, a variable separation (VS) strategy is presented to improve the dual decomposition and Lagrangian relaxation (DDLR) approach to propose a VSDDLR-SAA algorithm. Results of the numerical examples and a real-life case illustrate the effectiveness of the proposed model and VSDDLR-SAA algorithm. Comparison analysis of the 4PL network and the supply chain network shows that 4PL can deliver products within the prescribed time at a lower cost by cooperating with third-party logistics providers.

关键词： 4PL network design Service time Stochastic demand sample average approximation method Lagrangian relaxation

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A green intermodal service network design problem with travel time uncertainty

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TRANSPORTATION RESEARCH PART B-methodOLOGICAL 2016年第PartB期93卷 789-807页

作者： Demir, Emrah Burgholzer, Wolfgang Hrusovsky, Martin Arikan, Emel Jammernegg, Werner Van Woensel, Tom Eindhoven Univ Technol Sch Ind Engn Operat Planning Accounting & Control NL-5600 MB Eindhoven Netherlands WU Vienna Univ Econ & Business Inst Prod Management Welthandelspl 1 A-1020 Vienna Austria

In a more and more competitive and global world, freight transports have to overcome increasingly long distances while at the same time becoming more reliable. In addition, a raising awareness of the need for environmentally friendly solutions increases the importance of transportation modes other than road. Intermodal transportation, in that regard, allows for the combination of different modes in order to exploit their individual advantages. Inter modal transportation networks offer flexible, robust and environmentally friendly alternatives to transport high volumes of goods over long distances. In order to reflect these advantages, it is the challenge to develop models which both represent multiple modes and their characteristics (e.g., fixed-time schedules and routes) as well as the transhipment between these transportation modes. In this paper, we introduce a Green Intermodal Service Network Design Problem with Travel Time Uncertainty (GISND-TTU) for combined offline intermodal routing decisions of multiple commodities. The proposed stochastic approach allows for the generation of robust transportation plans according to different objectives (i.e., cost, time and greenhouse gas (GHG) emissions) by considering uncertainties in travel times as well as demands with the help of the sample average approximation method. The proposed methodology is applied to a real-world network, which shows the advantages of stochasticity in achieving robust transportation plans. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Stochastic service network design problem Intermodal transportation CO2-equivalent emissions Travel time uncertainty Demand uncertainty sample average approximation method

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Analysis of stochastic dual dynamic programming method

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第1期209卷 63-72页

作者： Shapiro, Alexander Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed sample average approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the "true" problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Stochastic programming Stochastic Dual Dynamic Programming algorithm sample average approximation method Monte Carlo sampling Risk averse optimization

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