On a Robust Approach for Stochastic Equilibrium Problems

随机规划在经济学、管理学、工程学、交通运输网络和环境等领域的最优决策建模中被运筹学家、经济学家和各种决策者/实践者广泛使用。当一个决策问题不仅涉及不确定性，而且涉及几个处于竞争关系中的决策者时，它就变成了一个随机博弈。理解这种博弈的一个重要方法是观察均衡结果。这些是比赛结束时的一组可能的结果，假设每个参与者都寻求优化自己的收益。随机规划和均衡中的一个基本问题涉及不确定性的表示。文献中的许多模型假定完全了解随机变量的分布(表示不确定性)。然而，在许多实际情况下，这样的分布并不是精确知道的，必须从历史数据估计或使用主观判断来构建。现有的信息往往不足以给出对所确定的分布的信心。在缺乏关于潜在分布的全部信息的情况下，仍有可能确定真实分布所在的一组可能的概率分布。对于该问题的稳健优化方法是基于在可能分布集合中的最差概率分布的情况下做出适当的决策，而随机均衡的稳健分析是在潜在随机元素的不完全信息的情况下考察最差均衡结果，稳健设计要求一个人设置适应任何最坏均衡结果的最优政策/参数。该项目被提议开发一个数学框架，该框架允许人们对具有关于潜在不确定性的不完全信息的随机均衡问题进行稳健分析，识别适应最坏可能均衡结果的最优政策/设计，开发有效的数值方法来求解新的数学模型，并将它们应用于一些有趣的经济和工程实际问题，特别是能源行业。

Stochastic programming has been extensively used by operation researchers, economists and various decision makers/practitioners to model optimal decision making in economics, management, engineering, transportation networks and the environment. When a decision problem involves not only uncertainty, but also severaldecision makers who are in a competitive relationship, it becomes a stochastic game. An important approach in understanding such a game is to look at the equilibrium outcomes. These are the set of possible outcomes atthe end of competition, given that each player seeks to optimize their own payoff.A fundamental issue in stochastic programming and equilibrium concerns the representation ofuncertainty. Many of the models in the literature assume complete knowledge of the distributions of random variables (representing the uncertainty). Inmany practical cases, however, such distributions are not known precisely and have to be either estimated from historical data or constructed usingsubjective judgements. The available information is often insufficient to give confidence in the distribution identified. In the absence of full information on the underlying distribution, it may still be possible toidentify a set of possible probability distributions within which the true distribution lies. While a robust optimization approach to this problem isbased on making the decision that would be appropriate given the worst probability distribution in the set of possible distributions, robust analysis of stochastic equilibrium is to look into worst equilibrium outcomes given the incomplete information of the underlying stochastic elements and robust designrequires one to set out optimal policy/parameters which accommodate any worst equilibrium outcomes.The project is proposed to develop a mathematical framework that allows one to carry out robust anaysis of a stochastic equilibrium problem with incomplete information on the underlying uncertainty, identify optimal policy/design which accommodate the worst possible equilibrium outcomes, develop efficient numerical methods for solving the new mathematical models and apply apply them to some interesting practical problems in economics and engineering with a particular focus on energy industry.