Optimization and Equilibria with Expectation Functions: Analysis, Inference and Sampling

数学优化和平衡问题在机器学习、统计学、经济和商业、卫生保健以及许多科学和工程分支中有着突出的应用。解决这些问题有助于了解复杂系统的性质和结构，并通过有效利用稀缺资源来更好地设计和控制这些系统。在每一个这样的问题的表述中都有许多参数。在许多情况下，由于缺乏完整的信息，特别是当这些参数描述未来事件时，某些参数的确切值是不可用的。在这种数据不确定性下，对复杂系统的长期行为进行管理的有效方法是在参数上引入概率分布，并在问题公式中使用期望函数。为了用期望函数在数值上解决这些问题，通常使用某些类型的近似。研究者研究了用期望函数定义的最优化和平衡问题的性质，这些问题与它们的近似之间的关系，以及解决这些问题的方法。特别令人感兴趣的是将这些思想应用于研究可再生能源和不可再生能源之间的电力市场竞争。这个项目的结果可以用来评估给定问题的适定性，测量从数值过程中得到的解的可靠性，并解决某些类型的问题。研究人员分析了由期望函数定义的优化和均衡问题的结构和性质，开发了推理程序和基于抽样的优化方法，并研究了在电力市场中的应用。他们的第一个目标是基于样本平均近似（SAA）问题的解，开发一种有效的推理方法来解决真正的优化或平衡问题。他们期望该方法适用于一般框架，该框架允许SAA函数是非光滑的，SAA解是不精确的，SAA渐近分布遵循分段正态结构，就像一般约束优化问题的情况一样。他们应用这种方法来预测SAA溶液的样本外性能。他们的第二个目标是重新审视现有的重要抽样技术，以有效地将它们整合到迭代优化算法中，以在某些设计参数下最小化罕见事件的概率，并研究概率函数的性质，并将原始问题的解与其凸替代品的解进行比较。研究者的第三个目标是研究电力市场中不同类型发电商之间竞争行为的随机均衡模型，并提供经典纳什-古诺均衡在支付函数非凹时的新推广。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical optimization and equilibrium problems have prominent applications in machine learning, statistics, economy and business, health care, and many branches of science and engineering. Solving these problems helps to gain knowledge about the nature and structure of complex systems, and to better design and control these systems by making efficient use of scarce resources. There are numerous parameters in the formulation of each such problem. In many cases the exact values of some parameters are not available due to the lack of complete information, especially when such parameters describe future events. An effective way to manage the long-term behaviors of complex systems under such data uncertainty is to introduce probability distributions on the parameters and use expectation functions in the problem formulations. To numerically solve those problems with expectation functions, certain types of approximations are commonly used. The investigators study properties of optimization and equilibrium problems defined with expectation functions, relations between these problems and their approximations, and methods to solve them. Of particular interest is application of these ideas to study the electricity market competition between renewable and nonrenewable energy sources. Results from this project can be used to evaluate the well-posedness of a given problem, measure the reliability of a solution obtained from a numerical procedure, and solve certain types of these problems.The investigators analyze the structure and properties of optimization and equilibrium problems defined by expectation functions, develop inference procedures and sampling-based optimization methods, and study an application to the electricity market. Their first goal is to develop an efficient inference method for the solution to the true optimization or equilibrium problem based on a solution to its sample average approximation (SAA) problem. They expect this method to work for a general framework that allows the SAA functions to be nonsmooth, the SAA solution to be inexact, and the SAA asymptotic distribution to follow a piecewise normal structure as in the cases of general constrained optimization problems. They apply this method to predict the out-of-sample performance of the SAA solutions. Their second goal is to revisit existing importance sampling techniques to efficiently incorporate them into an iterative optimization algorithm for minimizing the probability of a rare event under some design parameters, and study properties of the probability function and compare solutions of the original problem with those of its convex substitutes. The third goal of the investigators is to study a stochastic equilibrium model of the competition behavior between different types of energy generators in the electricity market and to provide a novel generalization of the classical Nash-Cournot equilibrium when the payoff functions are not concave.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.