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The policy graph decomposition of multistage stochastic programming problems

NETWORKS 2020年第1期76卷 3-23页

作者： Dowson, Oscar Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60208 USA

We propose the policy graph as a structured way of formulating a general class of multistage stochastic programming problems in a way that leads to a natural decomposition. We also propose an extension to the stochastic dual dynamic programming algorithm to solve a subset of problems formulated as a policy graph. This subset includes discrete-time, convex, infinite-horizon, multistage stochastic programming problems with continuous state and control variables. To demonstrate the utility of our algorithm, we solve an existing multistage stochastic programming problem from the literature based on pastoral dairy farming. We show that the finite-horizon model in the literature suffers from end-of-horizon effects, which we are able to overcome with an infinite-horizon model.

关键词： infinite horizon multistage policy graph stochastic programming

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Robust stochastic programming with uncertain probabilities

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IMA JOURNAL OF MANAGEMENT MATHEMATICS 2008年第3期19卷 289-321页

作者： Thiele, Aurelie Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA

stochastic programming has traditionally assumed the exact knowledge of the underlying scenario probabilities. In practice, however, such probabilities are difficult to estimate accurately and the optimal decision variables may be quite sensitive to the assumed distributions. This motivates the use of minimax stochastic models, where the decision maker minimizes the maximum expected cost over the set of possible probability distributions. We use ideas from the field of robust optimization to reformulate the minimax stochastic programming problem when probabilities belong to a polyhedral uncertainty set as a single convex problem, and show that it can be solved efficiently using the traditional techniques developed to address sequential decision making under uncertainty. In the two-stage setting, we describe how the Benders decomposition algorithm can be modified to solve the robust formulation. In the case of multiple stages, we build upon the recursive equations of dynamic programming to formulate an approach as tractable as the multi-stage stochastic problem where the probabilities are known exactly. Key contributions of this work are the following: (i) we show that the minimax approach is equivalent to the nominal stochastic programming problem with a penalty term, which measures the cost volatility due to the ambiguity on the probability estimates, and (ii) we provide deeper insights into the connection between the value of the recourse function in a given scenario and the worst-case probability associated with that outcome. The robust approach also allows the decision maker to adjust the parameters defining the uncertainty set in order to better capture his own trade-off between ambiguity and performance.

关键词： robust optimization uncertain probabilities uncertainty sets stochastic programming linear programming

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Duality and martingales: a stochastic programming perspective on contingent claims

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MATHEMATICAL programming 2002年第3期91卷 543-562页

作者： King, AJ IBM Corp Thomas J Watson Res Ctr Dept Math Sci IBM Res Div Yorktown Hts NY 10598 USA

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to die analysis of options pricing when modeling risk management concerns and the impact of spreads and mar.-in requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options, An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options, The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets.

关键词： options pricing martingales incomplete markets stochastic programming

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MOREAU ENVELOPE OF SUPREMUM FUNCTIONS WITH APPLICATIONS TO INFINITE AND stochastic programming\ast

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SIAM JOURNAL ON OPTIMIZATION 2021年第3期31卷 1635-1657页

作者： Perez-Aros, Pedro Vilches, Emilio Univ OHiggins Inst Ciencias Ingn Rancagua Chile

In this paper, we investigate the Moreau envelope of the supremum of a family of convex, proper, and lower semicontinuous functions. Under mild assumptions, we prove that the Moreau envelope of a supremum is the supremum of Moreau envelopes, which allows us to approximate possibly nonsmooth supremum functions by smooth functions that are also the suprema of functions. Consequently, we propose and study approximated optimization problems from infinite and stochastic programming for which we obtain zero-duality gap results and optimality conditions without the verification of constraint qualification conditions.

关键词： Moreau envelope subdifferential calculus supremum function infinite program-ming semi-infinite programming stochastic programming

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Multilevel Optimization Modeling for Risk-Averse stochastic programming

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INFORMS JOURNAL ON COMPUTING 2016年第1期28卷 112-128页

作者： Eckstein, Jonathan Eskandani, Deniz Fan, Jingnan Rutgers State Univ Dept Management Sci & Informat Syst Piscataway NJ 08854 USA Rutgers State Univ Rutgers Ctr Operat Res Piscataway NJ 08854 USA Rutgers State Univ Rutgers Business Sch Doctoral Program Management Sci Piscataway NJ 08854 USA Rutgers State Univ Doctoral Program Operat Res Piscataway NJ 08854 USA

Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. This paper summarizes the potential drawbacks of nested-form risk measure issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are time consistent even while using time-inconsistent risk measures and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant, and the associated models thus simplify to a more familiar single-objective form. Unfortunately, we also show that our proposed approach leads to NP-hard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard average value-at-risk and first-order mean-semideviation risk measures. Finally, we show that for a simple but reasonably realistic test application, the kind of models we propose, although drawn from an NP-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial mixed-integer linear programming solver. Therefore, there seems s

关键词： optimization modeling risk modeling coherent risk measures stochastic programming time consistency multilevel optimization bilevel programming computational complexity

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A two-stage interval-stochastic programming model for planning end-of-life vehicles allocation under uncertainty

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RESOURCES CONSERVATION AND RECYCLING 2015年 98卷 19-29页

作者： Simic, Vladimir Univ Belgrade Fac Transport & Traff Engn Belgrade 11000 Serbia

The ultimate fate of vehicles can vary, including being recycled, exported, abandoned, illegally treated in unauthorized facilities, stolen and processed for parts or illegally exported, etc. Nowadays, export of used vehicles represents the most significant barrier to the more efficient vehicle recycling in the EU, because millions of vehicles (mainly from Germany, Italy, UK and France), which are expected to go to domestic vehicle recycling factories are exported. As a result, how to allocate limited and frequently insufficient quantities of collected end-of-life vehicles (ELVs) to satisfy vast demands of vehicle recycling factories becomes a significant concern of many government or private authorities that control the ELV collection and treatment networks across the EU. In this paper, a two-stage interval-stochastic programming model is developed for supporting the management of ELV allocation under uncertainty. The formulated model can directly handle uncertainties expressed as either probability density functions or discrete intervals. Moreover, it can support the analysis of various policy scenarios that are associated with different levels of economic penalties when the promised ELV allocation targets are violated. The proposed model has been applied to a hypothetical case study in which several scenarios with different ELV allocation policies were comprehensively analyzed. The obtained results indicated that reasonable solutions had been generated. There is significant influence of parameter uncertainty on model solutions and the analyzed policy scenarios. Various policies in negotiating the ELV allocation targets with vehicle recycling factories would lead to different economic and risk values. Proposed model can serve as the support for authorities to identify ELV allocation targets that will secure maximized profits and minimized disruption risks of the vehicle recycling factories. It is applicable across vehicle recycling industry that processes dozens o

关键词： End-of-life vehicle Decision making stochastic programming Interval linear programming Uncertainty

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A two-stage stochastic programming model for the optimal design of distributed energy systems

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APPLIED ENERGY 2013年 103卷 135-144页

作者： Zhou, Zhe Zhang, Jianyun Liu, Pei Li, Zheng Georgiadis, Michael C. Pistikopoulos, Efstratios N. Tsinghua Univ Dept Thermal Engn State Key Lab Power Syst Beijing 100084 Peoples R China Aristotle Univ Thessaloniki Dept Chem Engn Thessaloniki 54124 Greece Univ London Imperial Coll Sci Technol & Med Dept Chem Engn CPSE London SW7 2AZ England

A distributed energy system is a multi-input and multi-output energy system with substantial energy, economic and environmental benefits. The optimal design of such a complex system under energy demand and supply uncertainty poses significant challenges in terms of both modelling and corresponding solution strategies. This paper proposes a two-stage stochastic programming model for the optimal design of distributed energy systems. A two-stage decomposition based solution strategy is used to solve the optimization problem with genetic algorithm performing the search on the first stage variables and a Monte Carlo method dealing with uncertainty in the second stage. The model is applied to the planning of a distributed energy system in a hotel. Detailed computational results are presented and compared with those generated by a deterministic model. The impacts of demand and supply uncertainty on the optimal design of distributed energy systems are systematically investigated using proposed modelling framework and solution approach. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： stochastic programming Distributed energy system Uncertainty Genetic algorithm The Monte Carlo method

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Scenario reduction in stochastic programming with respect to discrepancy distances

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2009年第1期43卷 67-93页

作者： Henrion, Rene Kuechler, Christian Roemisch, Werner Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany Humboldt Univ Inst Math D-10099 Berlin Germany

Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are a"not sign-discrepancies, where the class a"not sign of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to a"not sign-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.

关键词： stochastic programming Chance constraints Two-stage Mixed-integer Scenario reduction Discrepancy Kolmogorov metric

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A Bidding Strategy for Virtual Power Plants With the Intraday Demand Response Exchange Market Using the stochastic programming

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS 2018年第4期54卷 3044-3055页

作者： Hieu Trung Nguyen Le, Long Bao Wang, Zhaoyu Iowa State Univ Dept Elect & Comp Engn Ames IA 50011 USA Univ Quebec Inst Natl Rech Sci Energie Mat & Telecommun Montreal PQ H5A 1K6 Canada

This paper presents a mathematical model for the energy bidding problem of a virtual power plant (VPP) that participates in the regular electricity market and the intraday demand response exchange (DRX) market. Different system uncertainties due to the intermittent renewable energy sources, retail customers' demand, and electricity prices are considered in the model. The DRX market enables a VPP to purchase demand response services, which can be treated as "virtual energy resources," from several demand response providers to reduce the penalty cost on the deviation between the day-head bidding and the real-time dispatch. This could increase the expected profit and the renewable energy utilization of the VPP. The overall energy bidding problem is modeled as a three-stage stochastic program, which can be solved efficiently by the scenario-based optimization approach. Extensive numerical results show that the DRX market participation can improve the VPP's energy management.

关键词： Demand response exchange (DRX) energy bidding short-term electricity market stochastic programming virtual power plant (VPP)

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Day-ahead electric vehicle aggregator bidding strategy using stochastic programming in an uncertain reserve market

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IET GENERATION TRANSMISSION & DISTRIBUTION 2019年第12期13卷 2517-2525页

作者： Han, Bing Lu, Shaofeng Xue, Fei Jiang, Lin Xian Jiaotong Liverpool Univ Dept Elect & Elect Engn Suzhou 215123 Peoples R China Univ Liverpool Dept Elect Engn & Elect Liverpool L69 3GJ Merseyside England

Electric vehicle (EV) as dynamic energy storage systems could provide ancillary services to the grids. The aggregator could coordinate the charging/discharging of EV fleets to attend the electricity market to get profits. However, the aggregator profits are threatened by the uncertainty of the electricity market. In this study, an EV aggregator bidding strategy in the day-ahead market (DAM) is proposed, both reserve capacity and reserve deployment are considered. The novelty of this study is that: (i) The uncertainty of the reserve developments is addressed in terms of both time and amount. (ii) Scenario-based stochastic programming method is used to maximise the average aggregator profits based on one-year data. The proposed method, jointly considers the reserve capacity in the DAM and the reserve deployment requirements in the real-time market (RTM). (iii) The risk of the deployed reserve shortage is addressed by introducing a penalty factor in the model. (iv) An owner-aggregator contract is designed, which is used to mitigate the economic inconsistency issue between the EV owners and the aggregator. Results verify the performance of the proposed strategy, that is the average aggregator profits are guaranteed by maximising reserve deployment payments and mitigating the penalties in RTM and thus the reserve deployment requirements uncertainty is well managed.

关键词： power markets energy storage stochastic programming electric vehicles day-ahead electric vehicle aggregator bidding strategy uncertain reserve market dynamic energy storage systems ancillary services charging discharging electricity market EV aggregator day-ahead market DAM reserve capacity reserve developments Scenario-based stochastic programming method average aggregator profits real-time market deployed reserve shortage owner-aggregator contract EV owners reserve deployment payments reserve deployment requirements uncertainty

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