Collaborative Research: Successive Risk-Neutral Approximations of Dynamic Risk-Averse Optimization Problems

提出的研究旨在开发解决涉及风险规避偏好的随机动态优化问题的方法。风险厌恶的数学模型捕捉了随机结果的整个分布，增加了对小概率和高后果事件的关注。该项目将集中研究多阶段随机优化问题以及包含动态风险度量和动态随机排序约束的马尔可夫决策过程。所提出的数值方法将现代风险度量理论和随机阶数与大规模优化问题的分解技术、非光滑优化方法和随机控制方法相结合。该方法将基于风险厌恶问题的顺序风险中性近似。这些近似将用于设计具有动态风险度量和动态随机排序约束的多阶段问题的原始和对偶分解方法。我们将特别关注马尔可夫决策问题。一个理论的马尔可夫风险措施和风险厌恶动态规划将发展。风险规避动态规划的数值方法也将探讨顺序风险中性近似的思想。该项目将为高不确定性和高风险随机系统中涉及多阶段决策的领域提供定性进展。它将提供建模和算法工具，以形式化和解决长期规划问题，其中风险是一个重要问题，平均性能标准是不够的。这种性质的问题出现在供应链管理、军事规划问题、能源生产和分配、电信、保险和金融、医药等领域。该项目将有利于罗格斯大学和史蒂文斯理工学院的研究生教育。

The proposed research aims at developing methods for solving stochastic dynamic optimization problems that involve risk-averse preferences. Mathematical models of risk aversion capture entire distributions of random outcomes with increased attention to events of small probability and high consequences. The project will concentrate on multistage stochastic optimization problems and on Markov decision processes incorporating dynamic risk measures and dynamic stochastic ordering constraints. The proposed numerical approach integrates modern theories of risk measures and stochastic orders with decomposition techniques for large-scale optimization problems, methods of nonsmooth optimization, and stochastic control methods. The approach will be based on sequential risk-neutral approximations of risk-averse problems. The approximations will be used to devise primal and dual decomposition methods for multistage problems with dynamic risk measures and dynamic stochastic ordering constraints. Special attention will be paid to Markov decision problems. A theory of Markov risk measures and risk-averse dynamic programming will be developed. Numerical methods for risk-averse dynamic programming will also explore the idea of sequential risk-neutral approximations.The project will provide qualitative advance in areas involving multi-stage decision-making in stochastic systems under high uncertainty and risk. It will provide modeling and algorithmic tools to formalize and solve long-term planning problems in which risk is an important issue and average performance criteria are insufficient. Problems of this nature arise in supply chain management, military planning problems, energy production and distribution, telecommunication, insurance and finance, medicine, and other areas. The project will benefit the graduate education at Rutgers University and Stevens Institute of Technology.