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Risk minimization, regret minimization and progressive hedging algorithms

MATHEMATICAL PROGRAMMING 2020年第2期181卷 509-530页

作者： Sun, Jie Yang, Xinmin Yao, Qiang Zhang, Min Curtin Univ Fac Sci & Engn Perth WA Australia Natl Univ Singapore Sch Business Singapore Singapore Chongqing Normal Univ Sch Math Sci Chongqing Peoples R China East China Normal Univ Sch Stat Key Lab Adv Theory & Applicat Stat & Data Sci MOE Shanghai Peoples R China NYU Shanghai NYU ECNU Inst Math Sci Shanghai Peoples R China Chinese Acad Sci Xinjiang Inst Ecol & Geog State Key Lab Desert & Oasis Ecol Urumqi Peoples R China Univ Chinese Acad Sci Beijing Peoples R China

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.

关键词： progressive hedging algorithm Regret minimization Risk measures Stochastic optimization

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TWO-STAGE QUADRATIC GAMES UNDER UNCERTAINTY AND THEIR SOLUTION BY progressive hedging algorithmS

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SIAM JOURNAL ON OPTIMIZATION 2019年第3期29卷 1799-1818页

作者： Zhang, Min Sun, Jie Xu, Honglei Chinese Acad Sci Xinjiang Inst Ecol & Geog Urumqi 830011 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China Curtin Univ Sch EECMS Perth WA 6845 Australia

A model of a two-stage N-person noncooperative game under uncertainty is studied, in which at the first stage each player solves a quadratic program parameterized by other players' decisions and then at the second stage the player solves a recourse quadratic program parameterized by the realization of a random vector, the second-stage decisions of other players, and the first-stage decisions of all players. The problem of finding a Nash equilibrium of this game is shown to be equivalent to a stochastic linear complementarity problem. A linearly convergent progressive hedging algorithm is proposed for finding a Nash equilibrium if the resulting complementarity problem is monotone. For the nonmonotone case, it is shown that, as long as the complementarity problem satisfies an additional elicitability condition, the progressive hedging algorithm can be modified to find a local Nash equilibrium at a linear rate. The elicitability condition is reminiscent of the sufficient second-order optimality condition in nonlinear programming. Various numerical experiments indicate that the progressive hedging algorithms are efficient for mid-sized problems. In particular, the numerical results include a comparison with the best response method that is commonly adopted in the literature.

关键词： multistage noncooperative game under uncertainty progressive hedging algorithm stochastic linear complementarity problem stochastic variational inequality

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Analysis of the pavement deterioration uncertainty scenarios on pavement maintenance and rehabilitation planning optimization

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STRUCTURE AND INFRASTRUCTURE ENGINEERING 2025年第3期21卷 507-524页

作者： Fani, Amirhossein Golroo, Amir Fahmani, Mohammadsadegh Naseri, Hamed Nejad, Fereidoon Moghadas Amirkabir Univ Technol Dept Civil & Environm Engn Tehran Iran

This study aims to compare the results of stochastic and deterministic models and their corresponding optimal M&R actions considering different budget and deterioration rate uncertainty scenarios. Moreover, the impacts of uncertainty on network's scale are investigated. Therefore, this study applies Multi-Stage stochastic programming to model the uncertainty of these parameters in pavement M&R optimization. A new approach is proposed to investigate the effects of different uncertainty cases on the mentioned problem. Moreover, two pavement networks, including a large-scale and a small-scale, are utilized to evaluate the role of network size in the optimal solution to the pavement M&R optimization in the uncertainty conditions. Due to the high complexity of the large-scale M&R problem, progressive hedging algorithm as an effective decomposition technique is applied. Three different uncertainty cases, including low, medium, and high are considered for the deterioration rate. Furthermore, two scenarios are taken into account for the budget: low reduction and high reduction. The results show that the probability of selecting preventive maintenance in the optimal M&R plan is increased by increasing the severity of uncertainty cases. Therefore, preventive maintenance is the most effective pavement treatment to reduce the adverse effects of budget and pavement deterioration uncertainty.

关键词： Maintenance and rehabilitation uncertainty risk pavement deterioration stochastic programming progressive hedging algorithm sensitivity analysis

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A multi-stage stochastic programming model for multi-mission selective maintenance optimization

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2025年 254卷

作者： Ghorbani, Milad Nourelfath, Mustapha Gendreau, Michel Interuniv Res Ctr Enterprise Networks Logist & Tra Quebec City PQ Canada Laval Univ Fac Sci & Engn Dept Mech Engn Quebec City PQ Canada Polytech Sch Montreal Dept Math & Ind Engn Montreal PQ Canada

This research introduces a novel selective maintenance model in the case of systems undergoing multiple consecutive missions. The model considers uncertainties related to future operating conditions during each mission. Within each maintenance break, various optional actions ranging from replacements which are perfect to imperfect and also minimal repairs can be chosen for individual components. Evaluating the probabilities of successful future mission accounts for uncertainties associated with component operational conditions. The selective maintenance problem is formulated as a nonlinear mixed-integer model for optimization, and computational challenges are addressed using the progressive hedging algorithm. Numerical examples validate the new proposed model and illustrate the benefits of the model by estimating a more realistic reliability level and lower maintenance cost.

关键词： Selective maintenance Stochastic programming Uncertain operating conditions progressive hedging algorithm

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Optimal Configuration of Mobile-Stationary Hybrid Energy Storage Considering Seismic Hazards

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ENERGIES 2025年第8期18卷 2052-2052页

作者： Deng, Chengcheng Shen, Xiaodong Tang, Xisheng Sichuan Univ Sch Elect Engn Chengdu 610065 Peoples R China Chinese Acad Sci Inst Elect Engn Beijing 100190 Peoples R China

The occurrence of extreme disasters, such as seismic hazards, can significantly disrupt transportation and distribution networks (DNs), consequently impacting the post-disaster recovery process. Restoring load using distributed generation represents an important approach to improving the resilience of DNs. However, using these resources to provide resilience is not enough to justify having them installed economically. Therefore, this paper proposes a two-stage stochastic mixed-integer programming (SMIP) model for the configuration of stationary energy storage systems (SESSs) and mobile energy storage systems (MESSs) during earthquakes. The proposed model comprehensively considers both normal and disaster operation scenarios of DNs, maximizing the grid's economic efficiency and security. The first stage is to make decisions about the location and size of energy storage, using a hybrid configuration scheme of second-life batteries (SLBs) for SESSs and fresh batteries for MESSs. In the second stage, the operating costs of DNs are evaluated by minimizing normal operating costs and reducing load loss during seismic events. Additionally, this paper proposes a scenario reduction method based on hierarchical sampling and distance reduction to generate representative fault scenarios under varying earthquake magnitudes. Finally, the progressive hedging algorithm (PHA) is employed to solve the model. The case studies of the IEEE 33-bus and 12-node transportation network are conducted to validate the effectiveness of the proposed method.

关键词： seismic hazards mobile energy storage second-life batteries two-stage stochastic programming progressive hedging algorithm

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Minimizing buffered probability of exceedance by progressive hedging

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MATHEMATICAL PROGRAMMING 2020年第2期181卷 453-472页

作者： Rockafellar, R. Tyrrell Uryasev, Stan Univ Washington Dept Math Box 354350 Seattle WA 98195 USA Univ Florida Dept Syst & Ind Engn Gainesville FL 32611 USA

Stochastic programming problems have for a long time been posed in terms of minimizing the expected value of a random variable influenced by decision variables, but alternative objectives can also be considered, such as minimizing a measure of risk. Here something different is introduced: minimizing the buffered probability of exceedance for a specified loss threshold. The buffered version of the traditional concept of probability of exceedance has recently been developed with many attractive properties that are conducive to successful optimization, in contrast to the usual concept, which is often posed simply as the probability of failure. The main contribution here is to demonstrate that in minimizing buffered probability of exceedance the underlying convexities in a stochastic programming problem can be maintained and the progressive hedging algorithm can be employed to compute a solution.

关键词： Convex stochastic programming problems Probability of failure Probability of exceedance Buffered probability of failure Buffered probability of exceedance Quantiles Superquantiles Conditional value-at-risk progressive hedging algorithm

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Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

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MATHEMATICAL PROGRAMMING 2019年第1-2期174卷 453-471页

作者： Rockafellar, R. Tyrrell Sun, Jie Univ Washington Dept Math Box 354350 Seattle WA 98195 USA Curtin Univ Dept Math & Stat Bentley WA 6102 Australia

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.

关键词： progressive hedging algorithm Stochastic variational inequality problems Stochastic complementarity problems Stochastic programming problems Maximal monotone mappings Proximal point algorithm Problem decomposition

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Adaptive monitoring of the progressive hedging penalty for reservoir systems management

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ENERGY SYSTEMS-OPTIMIZATION MODELING SIMULATION AND ECONOMIC ASPECTS 2014年第2期5卷 307-322页

作者： Zephyr, Luckny Lang, Pascal Lamond, Bernard F. Univ Laval Operat & Decis Syst Dept Laval PQ Canada

Reservoir systems operations problems are in essence stochastic because of the uncertain nature of natural inflows. This leads to very large stochastic models that may not be easy to handle numerically. In this paper, we revisit the decomposition method developed by Rockafellar and Wets (Math Oper Res 119-147, 1991) by proposing new heuristics to initialize and dynamically adjust the penalty parameter of the augmented Lagrangian function on which this method is based. The heuristics are tested on multi-reservoir problems generated randomly and compared with the traditional strategy of setting the penalty parameter to a fixed value.

关键词： Reservoir systems operations Stochastic programming models Decomposition method Heuristic progressive hedging algorithm

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progressive hedging and Sample Average Approximation for the Two-stage Stochastic Traveling Salesman Problem 6

Progressive Hedging and Sample Average Approximation for the...

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6th International Conference on Operations Research and Enterprise Systems (ICORES)

作者： Adasme, Pablo Leung, Janny Soto, Ismael Univ Santiago Chile Dept Ingn Elect Ave Ecuador 3519 Santiago Chile Chinese Univ Hong Kong Shaw Coll 2001 Longxiang Blvd Shenzhen Peoples R China

ISBN: (纸本)9789897582189

In this paper, we propose an adapted version of the progressive hedging algorithm (PHA) (Rockafellar and Wets, 1991;Lokketangen and Woodruff, 1996;Watson and Woodruff, 2011) for the two-stage stochastic traveling salesman problem (STSP) introduced in (Adasme et al., 2016). Thus, we compute feasible solutions for small, medium and large size instances of the problem. Additionally, we compare the PHA method with the sample average approximation (SAA) method for all the randomly generated instances and compute statistical lower and upper bounds. For this purpose, we use the compact polynomial formulation extended from (Miller et al., 1960) in (Adasme et al., 2016) as it is the one that allows us to solve large size instances of the problem in short CPU time with CPLEX. Our preliminary numerical results show that the results obtained with the PHA algorithm are tight when compared to the optimal solutions of small and medium size instances. Moreover, we obtain significantly better feasible solutions than CPLEX for large size instances with up to 100 nodes and 10 scenarios in significantly low CPU time. Finally, the bounds obtained with SAA method provide an average reference interval for the stochastic problem.

关键词： Two-stage Stochastic Programming Traveling Salesman Problem progressive hedging algorithm Sample Average Approximation Method

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Short-term Transmission Maintenance Scheduling Considering Network Topology Optimization

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Journal of Modern Power Systems and Clean Energy 2022年第4期10卷 883-893页

作者： Weixin Zhang Bo Hu Kaigui Xie Changzheng Shao Tao Niu Jiahao Yan Lvbin Peng Maosen Cao Yue Sun State Key Laboratory of Power Transmission Equipment&System Security and New Technology at Chongqing University Chongqing 400044China

With the increasing penetration of renewable energy sources,transmission maintenance scheduling(TMS)will have a larger impact on the accommodation of wind ***,the more flexible transmission network topology owing to the network topology optimization(NTO)technique can ensure the secure and economic operation of power *** paper proposes a TMS model considering NTO to decrease the wind curtailment without adding control *** problem is formulated as a two-stage stochastic mixed-integer programming *** first stage arranges the maintenance periods of transmission *** second stage optimizes the transmission network topology to minimize the maintenance cost and system operation in different wind speed *** proposed model cannot be solved efficiently with off-theshelf solvers due to the binary variables in both ***,the progressive hedging algorithm is *** results on the modified IEEE RTS-79 system show that the proposed method can reduce the negative impact of transmission maintenance on wind accommodation by 65.49%,which proves its effectiveness.

关键词： Mixed-integer linear programming network topology optimization progressive hedging algorithm stochastic optimization transmission maintenance scheduling wind curtailment

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