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

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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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 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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Multi-period time window assignment for attended home delivery

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第1期316卷 295-309页

作者： Cote, Jean-Francois Mansini, Renata Raffaele, Alice Univ Laval CIRRELT 2325 Rue Terrasse Quebec City PQ G1V 0A6 Canada Univ Brescia Dept Informat Engn Via Branze 38 I-25123 Brescia Italy Univ Calabria Dept Mech Energy & Management Engn Via Pietro Bucci Cubo 46-C I-87036 Arcavacata Di Rende Italy

We study a multi -period stochastic variant of the Time Window Assignment Vehicle Routing Problem, where customers' demands, locations, and service times are uncertain. Customers are partitioned into geographical zones, each of which has to be visited a predetermined number of times over a planning period of several days. Whenever a zone is visited, a time window is assigned. Time windows are decided before knowing customers and their demands. A fleet of homogeneous vehicles is available to serve customers each day. At a tactical level, the problem looks for a static time window assignment that minimizes the expected traveling costs plus the expected penalty costs for unserved customers. We propose a two -stage formulation and a solution approach, which relies on the sample average approximation method, while encompassing a perturbation method to assign time windows in the first stage and an Adaptive Large Neighborhood Search to optimize routes in the second stage. We experimentally evaluate three instance sets, including real ones from a Canadian company, comparing our results to lower bounds from the exact solution of a deterministic equivalent formulation over a finite number of scenarios. Our method outperforms the manual approach used by the company.

关键词： Transportation Time window assignment Vehicle routing problem Stochastic programming 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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Distributionally robust chance constrained planning model for energy storage plants based on Kullback-Leibler divergence

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ENERGY REPORTS 2021年 7卷 5203-5213页

作者： Le, Jian Liao, Xiaobing Zhang, Lina Mao, Tao Wuhan Univ Sch Elect Engn & Automat Wuhan 430072 Peoples R China Wuhan Inst Technol Sch Elect & Informat Engn Wuhan 430205 Peoples R China State Grid Shanxi Elect Power Corp Taiyuan 030001 Shanxi Peoples R China Wuhan Donghu Univ Wuhan 430074 Peoples R China

Nowadays, the high penetration of renewable energy, with variable and unpredictable nature, poses major challenges to operation and planning studies of power systems. Employing energy storage plants has been introduced as an effective solution to alleviate these challenges. Several studies have been presented in the literature to provide a framework for planning studies of energy storage plants. However, they usually have two main drawbacks: (i) ignoring the lifetime varying characteristic of energy storage, (ii) inability to model the charge/discharge schedule of energy storage accurately. This paper amends the abovementioned shortcomings by proposing a distributionally robust planning method based on Kullback-Leibler divergence. According to the power function of lifespan of electrochemical energy storage, the lifespan model of energy storage plants with equivalent full cycles times is established. Considering the lifespan model constraints of the energy storage plant and system operating constraints, the planning model of energy storage plants is constructed with the lifespan cycle cost and units' operation cost as the objective. Furthermore, the ambiguity set of wind farm output based on Kullback-Leibler divergence is embedded into the planning model of energy storage plants, and the distributionally robust planning model of energy storage plant is transformed into mixed integer linear programming model by sample average approximation method. A modified IEEE-30 bus system with two wind farms is studied to demonstrate the effectiveness. The results show distributionally robust planning model reduces about 10% planning cost compared with the traditional robust planning model, and increases the energy storage capacity by 12MW compared with the stochastic planning method. Other, the planning cost of energy storage plants will decrease by 5% similar to 23% when the wind abandoned rate increases by 2% and decreases slowly with the increase of abandoned wind rate. (C)

关键词： Electrochemical energy storage Ambiguity set Distributionally robust planning sample average approximation method Mixed integer linear programming

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Fourth-party logistics network design under uncertainty environment

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COMPUTERS & INDUSTRIAL ENGINEERING 2022年 167卷

作者： Yin, Mingqiang Huang, Min Wang, Xingwei Lee, Loo Hay Northeastern Univ Coll Informat Sci & Engn State Key Lab Synthet Automat Proc Ind Shenyang 110819 Liaoning 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

A novel fourth-party logistics (4PL) network design problem under uncertainty environment is studied in the current work. Demand uncertainty and two types of disruptions, facility and third-party logistics (3PL) disruptions, are simultaneously considered. To minimize the total cost, a two-stage stochastic programming model is presented, which is further approximated as a mixed-integer linear programming model using the sample average approximation (SAA) method. As a large number of disruption and demand scenarios generated in the approximation process lead to challenges in model solving, the scenario reduction (SR), SAA, dual decomposition and Lagrangian relaxation (DDLR) approaches are integrated to present an SR-DDLRSAA algorithm. The effectiveness of our model and algorithm is illustrated by adopting numerical instances and real-life cases. A comparative analysis indicates that the impact of disruptions on the 4PL network is closely related to the values of the disruption probability and the fluctuation degree of uncertain demand. Furthermore, we extend the proposed basic model by considering demand uncertainty and three types of disruptions, namely supply, facility, and 3PL disruptions, and analyze the impact of supply disruption on the 4PL network.

关键词： 4PL network design Uncertain demand Disruptions sample average approximation method Scenario reduction approach

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ON FEASIBILITY OF sample average approximation SOLUTIONS

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SIAM JOURNAL ON OPTIMIZATION 2020年第3期30卷 2026-2052页

作者： Liu, Rui Peng Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

When there are infinitely many scenarios, the current studies of two-stage stochastic programming problems rely on the relatively complete recourse assumption. However, such an assumption can be unrealistic for many real-world problems. This motivates us to study general stochastic programming problems where the sample average approximation (SAA) solutions are not necessarily feasible. When the problems are convex and the true solutions lie in the interior of feasible solutions, we show that the portion of infeasible SAA solutions decays exponentially as the sample size increases. We also study functions with chain-constrained domain and show that the portion of SAA solutions having a low degree of feasibility decays exponentially as the sample size increases. This result is then extended to multistage stochastic programming.

关键词： (multistage) stochastic programming sample average approximation method feasibility convergenceexponential rate

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

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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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On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

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MATHEMATICAL PROGRAMMING 2017年第1期165卷 391-431页

作者： Yousefian, Farzad Nedic, Angelia Shanbhag, Uday V. Oklahoma State Univ Sch Ind Engn & Management Stillwater OK 74074 USA Arizona State Univ Sch Elect Comp & Energy Engn Tempe AZ 85287 USA Penn State Univ Ind & Mfg Engn State Coll PA 16802 USA

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594-609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA, in which we employ a weighted iterate-averaging, parameterized by a scalar r where provides us with the standard averaging scheme. The main contributions are threefold: (i) when and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate after k steps;(ii) when , and smoothing and regularization are suppressed, the gap function admits the rate , thus improving the rate under standard averaging;and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for . Notably, we prove the superiority of the scheme with with its counterpart with in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash-Cournot game with merely monotone and non-Lipschitzian maps.

关键词： STOCHASTIC approximation REGULARIZATION parameter MONOTONE operators TIKHONOV regularization sample average approximation method

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