Exploiting Structure in Nonsmooth Optimization Problems

在非光滑优化问题中利用结构

• 批准号: 355571-2013
• 负责人: Hare, Warren
• 金额: $ 1.75万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=636726
• 关键词: Exploiting; Structure; Nonsmooth; Optimization; Problems

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.

优化，最小化或最大化函数的研究，是现代社会的基本。应用非常多样化，包括以下问题：在设计新建道路时最小化建筑成本，最大化新微芯片设计的效率（速度），最小化预期的医院病床拥挤，以及最大化地震阻尼器在建筑物改造中的有效性。尽管计算能力呈指数级增长，算法设计也有显著改进，但现代优化应用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的研究和知识。在优化应用中，我们专注于现实世界的问题。我们开发方法来检测问题的数学公式中的结构。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。一旦检测到，我们将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，我们开发了新的方法来解决模拟模型中出现的优化问题。模拟的优化是特别具有挑战性的，因为目标函数是一个“黑盒子”，不能分析探索。我们寻求方法来检测和利用黑盒内的结构，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，这项研究将提供宝贵的资产，研究人员在学科的优化传统上没有涉及。在最优化理论中，我们探索了“正则”和“参数正则”函数的概念。这些概念提供了凸性的扩展，使得可以有效地处理更广泛的函数类。