Risk Averse Multistage Stochastic Integer Programming

在各种应用中，多阶段随机规划已成为求解不确定条件下序列决策的重要框架。这种方法背后的数学模型构成了一类极具挑战性的优化问题。在这些问题的解决策略方面已经取得了非常重大的研究进展，但其中大部分研究都局限于风险中性设置和底层数学结构简单（凸）的情况。最近，在电力系统应用中，可再生能源的渗透及其结构复杂性所带来的日益增加的波动性激发了该框架中对风险和非凸结构的明确考虑。该项目旨在为规避风险的多阶段随机规划做出基础理论和算法贡献，特别是用整数变量来模拟非凸性，并研究其在能源部门的应用。如果成功，该项目的结果将为电力系统运营商提供有价值的规划和调度工具。这些发展也会影响到其他各种应用领域，包括制造业、金融和服务业。该项目的结果将通过出版物和会议报告传播，并将在随机规划的研究生课程中采用。该项目将通过支持博士生的研究，为培养未来的学者和研究人员做出贡献。该项目将为规避风险的多阶段随机整数规划开发基于抽样和动态规划的方法。支持这些方法的理论和算法在风险中性和线性设置中得到了广泛的研究。在这个框架中纳入风险规避提出了关于风险度量的关键问题，这些风险度量从动态决策的角度来看是有意义的，并且从通过抽样逼近和通过动态规划优化的角度来看，在计算上是有吸引力的。整合整数决策引入了非凸性，并且需要新的分析和算法技术来解决它们。研究将集中在假设阶段独立的多阶段问题，或者更一般地说，不确定数据过程和二元状态变量的马尔可夫结构，并利用所得结构开发可扩展的方法。特别是，该项目将调查(i)各种风险措施的建模和结构问题，（ii）基于抽样的评估和优化风险规避目标的方法，以及（iii）近似动态规划方法。

Multistage stochastic programming is well established as an important framework for sequential decision making under uncertainty in a variety of applications. The mathematical models underlying this approach constitute an extremely challenging class of optimization problems. There has been very significant research progress in solution strategies for these problems, but much of it has been restricted to the risk neutral setting and when the underlying mathematical structures are simple (convex). Recently, in power systems applications, the increasing volatility brought forth by the penetration of renewable energy sources along with their structural complexities motivate the explicit consideration of risk and non-convex structures in this framework. This project aims to make fundamental theoretical and algorithmic contributions to risk-averse multistage stochastic programming, especially with integer variables to model non-convexities, and investigate its applications in the energy sector. If successful, results from this project will provide valuable planning and scheduling tools for power system operators. The developments can also impact a variety of other application areas including manufacturing, finance and service. The results of this project will be disseminated through publications and conference presentations, and will be adopted in graduate courses on stochastic programming. The project will contribute to the training of future academics and researchers by supporting the research of doctoral students.The project will develop sampling and dynamic programming based approaches for risk-averse multistage stochastic integer programs. Theory and algorithms underpinning such approaches have been researched extensively in the risk neutral and linear setting. Incorporating risk aversion in this framework raises crucial questions regarding risk measures that make sense from the point of view of dynamic decisions, and are computationally attractive from the point of view of approximation via sampling and optimization via dynamic programming. Incorporating integer decisions introduces nonconvexities, and requires novel analysis and algorithmic techniques for their resolution. The research will focus on multistage problems under the assumption of stagewise independent, or more generally Markovian, structure of the uncertain data process and binary state variables, and exploit the resulting structure to develop scalable approaches. In particular, the project will investigate (i) modeling and structural issues for various risk measures, (ii) sampling based approaches for evaluating and optimizing risk-averse objectives, and (iii) approximate dynamic programming approaches.