Exploiting Structure in Nonsmooth Optimization Problems

优化，即研究函数的最小化或最大化，在现代社会中是必不可少的。应用非常多样化，包括这样的问题：在设计新道路时最大限度地降低建筑成本，最大限度地提高新的微芯片设计的效率（速度），最大限度地减少预期的医院床位拥堵，以及在建筑改造中最大限度地提高地震阻尼器的有效性。尽管计算能力呈指数级增长，算法设计也有了重大改进，但现代优化应用比以往任何时候都更加复杂。在本提案中，我们推进优化应用，算法设计和优化理论的研究和知识。在优化应用中，我们关注现实世界的问题。我们开发的方法来检测结构在一个问题的数学公式。有些结构是自然产生的，例如在有限极大极小问题中，而其他结构可以通过应用某些建模和优化技术（例如正则化）来创建。一旦发现，我们寻求利用结构，以提高解决的时间和质量。在算法设计中，我们开发了解决仿真模型中出现的优化问题的新方法。优化模拟尤其具有挑战性，因为目标函数是一个无法分析探索的“黑盒子”。我们寻求方法来检测和利用黑箱中的结构，以提高收敛速度和解决方案的质量。随着仿真软件成为越来越普遍的创新工具，这项研究将为传统上没有涉及优化的学科的研究人员提供宝贵的资产。在优化理论中，我们探讨了“准正则”和“参数准正则”函数的概念。这些概念提供了凸性的扩展，使其能够有效地处理更广泛的函数类。

Optimization, the study of minimizing or maximizing a function, is essential in modern society. Applications are extremely diverse and include problems like: minimizing the construction costs when designing new roads, maximizing the efficiency (speed) of a new microchip design, minimizing expected hospital bed congestion, and maximizing the effectiveness of seismic dampers in building retrofitting. Despite the exponential growth in computing power and significant improvements in algorithm designs, modern applications of optimization are more complex than ever. In this proposal, we advance research and knowledge in Optimization Applications, Algorithm Design, and Optimization Theory. In Optimization Applications, we focus on real-world problems. We develop methods to detect structures within a problem's mathematical formulation. Some structures arise naturally, such as in finite minimax problems, while other structures can be created by the application of certain modelling and optimization techniques, such as regularization. Once detected, we seek to exploit the structures to improve solution time and quality. In Algorithm Design, we develop new methods for solving optimization problems that arise from simulation models. Optimization of simulations is particularly challenging, as the objective function is a "black-box" that cannot be explored analytically. We seek methods to detect and exploit structures within the black-box to improve convergence rates and solution quality. As simulation software becomes an increasingly prevalent tool for innovation, this research will provide valuable assets to researchers in disciplines where optimization has not traditionally been involved. In Optimization Theory, we explore the notions of "prox-regular" and "parametrically prox-regular" functions. These notions provide an extension of convexity that make it possible to work effectively with a much broader class of functions.