Exploiting Structure in Nonsmooth Optimization Problems

最优化是对函数最小化或最大化的研究，在现代社会中是必不可少的。应用非常多样化，包括以下问题：在设计新道路时将建筑成本降至最低、最大限度地提高新微芯片设计的效率(速度)、最大限度地减少预期的医院床位拥堵，以及最大限度地提高建筑物改造中抗震阻尼器的有效性。尽管计算能力呈指数级增长，算法设计也有了显著改进，但现代优化应用程序比以往任何时候都更加复杂。在这个提案中，我们推进了优化应用、算法设计和优化理论方面的研究和知识。 在优化应用中，我们关注的是现实世界中的问题。我们开发方法来检测问题的数学公式中的结构。一些结构是自然产生的，例如在有限极大极小问题中，而另一些结构可以通过应用某些建模和优化技术来创建，例如正则化。一旦检测到，我们就寻求利用这些结构来改善解决方案的时间和质量。 在算法设计中，我们开发了解决模拟模型中出现的优化问题的新方法。模拟的优化尤其具有挑战性，因为目标函数是一个无法用分析方法探索的“黑箱”。我们寻求检测和利用黑盒内部结构的方法来提高收敛速度和解的质量。随着模拟软件成为越来越普遍的创新工具，这项研究将为传统上不涉及优化的学科的研究人员提供宝贵的资产。 在最优化理论中，我们探索了“参数正则”和“参数正则”函数的概念。这些概念提供了凸性的扩展，使其能够有效地与更广泛的函数类一起工作。

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.