Risk Averse Multistage Stochastic Integer Programming

多阶段随机规划作为不确定条件下序贯决策的一个重要框架，在各种应用中都得到了很好的应用。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有了非常重要的研究进展，但大部分都局限在风险中性的设置和基本数学结构简单(凸)的情况下。近年来，在电力系统应用中，可再生能源的普及带来了越来越大的波动性，以及其结构的复杂性，促使在该框架中明确考虑风险和非凸结构。本项目旨在为风险厌恶多阶段随机规划，特别是用整数变量来模拟非凸性做出基本的理论和算法贡献，并研究其在能源领域的应用。如果成功，该项目的成果将为电力系统运营商提供有价值的计划和调度工具。这些发展还可能影响到包括制造业、金融业和服务业在内的各种其他应用领域。这一项目的成果将通过出版物和会议报告加以传播，并将在随机规划研究生班中采用。该项目将通过支持博士生的研究，为未来学者和研究人员的培训做出贡献。该项目将开发基于抽样和动态规划的方法，用于风险厌恶的多阶段随机整数规划。支撑这些方法的理论和算法已经在风险中性和线性环境下得到了广泛的研究。将风险规避纳入这个框架提出了有关风险度量的关键问题，这些度量从动态决策的角度来看是有意义的，从通过抽样进行近似和通过动态规划进行优化的角度来看在计算上是有吸引力的。合并整数判决引入了非凸性，并需要新的分析和算法技术来解决它们。研究将集中在不确定数据过程的阶段性独立结构或更一般的马尔可夫结构和二元状态变量的假设下的多阶段问题，并利用所得到的结构来开发可扩展的方法。特别是，该项目将研究(1)各种风险衡量的建模和结构问题，(2)评估和优化避险目标的基于抽样的方法，以及(3)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.