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ON FEASIBILITY OF sample average approximation SOLUTIONS

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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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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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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Chance-Constrained Programming method of IT Risk Countermeasures for Social Consensus Making

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2015年第5期45卷 725-733页

作者： Samejima, Masaki Sasaki, Ryoichi Osaka Univ Grad Sch Informat Sci & Technol Dept Multimedia Engn Suita Osaka 5650871 Japan Tokyo Denki Univ Sch Sci & Technol Future Life Tokyo 1208551 Japan

The authors address a social consensus making support in discussing countermeasures for information technology risks (IT risks). For supporting stakeholders' discussion on which IT risk countermeasures the stakeholders should implement, experts of the risk management estimate parameter values of the countermeasure, define a goal and constraints, and formulate the decision problem of the countermeasures to be implemented as one of 0-1 integer programming problems. Because parameter values and constraint values are uncertain, the decision problem is reformulated as a chance-constrained programming problem. The sample average approximation method is a well-known method for solving the chance-constrained programming problem. However, the computational time is still so long that the opinion leaders cannot use a solution of the chance-constrained programming problem in their discussion. The authors propose a high-speed chance-constrained programming method by aggregating the constraints that are generated by approximation of the problem in the sample average method. By applying the proposed method to real decision problems, the authors confirmed that computational time is decreased to 1 min while obtaining the same error rate and the same rate of the feasible solutions as a conventional method.

关键词： Chance-constrained programming constraint aggregation information technology risk (IT risk) sample average approximation method social consensus making

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ROBUST STOCHASTIC approximation APPROACH TO STOCHASTIC PROGRAMMING

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SIAM JOURNAL ON OPTIMIZATION 2009年第4期19卷 1574-1609页

作者： Nemirovski, A. Juditsky, A. Lan, G. Shapiro, A. Georgia Inst Technol Atlanta GA 30332 USA Univ Grenoble 1 F-38041 Grenoble 9 France

In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte Carlo sampling techniques, namely, the stochastic approximation (SA) and the sample average approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say, linear) structure of the considered problem, while the SA approach is a crude subgradient method, which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convex-concave stochastic saddle point problems and present (in our opinion highly encouraging) results of numerical experiments.

关键词： stochastic approximation sample average approximation method stochastic programming Monte Carlo sampling complexity saddle point minimax problems mirror descent algorithm

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Minimum mean-squared deviation method for stochastic complementarity problems

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2016年第7期93卷 1173-1187页

作者： Yu, Haodong Shanghai Lixin Univ Commerce Sch Math & Informat Shanghai Peoples R China

In this paper, we propose a new reformulation for stochastic complementarity problems (SCPs). The new formulation is based on the minimum mean-squared deviation rule in statistics. Under mild conditions, we prove the existence of the solution of the new reformulation for SCP. Furthermore, we present a smoothing sample average approximation method for solving the problems. The convergence properties of the optimal solutions of the approximation problems are studied under mild conditions. Finally, some numerical results are listed as well.

关键词： stochastic complementarity problems minimum mean-squared deviation sample average approximation method complementarity function smoothing technique

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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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Stochastic methods Based on VU-Decomposition methods for Stochastic Convex Minimax Problems

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MATHEMATICAL PROBLEMS IN ENGINEERING 2014年第1期2014卷 1-5页

作者： Lu, Yuan Wang, Wei Chen, Shuang Huang, Ming Shenyang Univ Normal Coll Shenyang 110044 Peoples R China Liaoning Normal Univ Sch Math Dalian 116029 Peoples R China Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China

This paper applies sample average approximation (SAA) method based on VU-space decomposition theory to solve stochastic convex minimax problems. Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one and convergence is exponentially fast with the increase of sample size. Based on the VU-theory, a superlinear convergent VU-algorithm frame is designed to solve the SAA problem.

关键词： DECOMPOSITION method (Mathematics) STOCHASTIC analysis sample average approximation method STOCHASTIC convergence PROBABILITY theory sample size (Statistics)

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Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第2期162卷 405-427页

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

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the sample average approximation method in order to approach this problem. We prove that the Hausdorff-Pompeiu distance between the weakly Pareto sets associated with the sample average approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the sample average approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the sample average approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.

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

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Stochastic programming approach to optimization under uncertainty

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MATHEMATICAL PROGRAMMING 2008年第1期112卷 183-220页

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

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages. We discuss an extension of coherent risk measures to a multistage setting and, in particular, dynamic programming equations for such problems.

关键词： two and multistage stochastic programming complexity Monte Carlo sampling sample average approximation method coherent risk measures dynamic programming conditional risk mappings

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