Applied Variational Analysis: Theory, Algorithms, and Applications

应用变分分析：理论、算法和应用

• 批准号: RGPIN-2017-06642
• 负责人: Wang, Shawn
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679554
• 关键词: Applied; Variational; Analysis; Theory; Algorithms

My research is in applied variational analysis. Variational analysis is an extension of convex analysis and classical analysis to encompass a variety of nondifferential functions and mappings. With its tools well-developed, seeking their applications in different areas, practical problems, and computation algorithms are essential. ******One important result for set-valued mappings is the Attouch-Thera duality, which assumes that the primal has a solution. If the primal has no solution, it is not clear whether an analogue of the Fenchel-Rockafellar duality for maximal monotone mappings exists. Epi-convergence is essential for studying convergence of extended-real-valued functions. In this topology, what can we say about minimization behaviors of nonconvex functions? Proximal average of convex functions is a powerful tool in modern convex analysis, but its generalizations to nonconvex functions are much less explored. Up to now, almost all splitting algorithms need both operators to be monotone. What happens if at least one operator is not monotone? Prox-regularity of functions and metric regularity of mappings have been systematically studied by Poliquin, Rockafellar, Thibault, Mordukhovich, Ioffe, Lewis, et al., but their algorithmic consequences await a comprehensive study. From convex functions to nonconvex functions, monotone mappings to nonmonotone mappings, these generalizations are intrinsically hard, often they require new methodologies. Although some progress has been made, it is not satisfactory at all.******In this proposal, I plan to (1) study Attouch-Thera's duality, generic minimization properties of nonconvex functions by envelopes, proximal average applications and extensions, and the second order nonsmooth analysis of envelopes by Mordukhovich's coderivative analysis and Rockafellar's proto-derivative analysis; (2) develop algorithms and a local convergence theory for solving zeros of a sum of nonmonotone mappings, and for minimizing a sum of nonconvex functions. Works by Chen and Rockafellar, Bauschke, Combettes, Noll, and Thera will be examined. Generalized local nonexpansive mappings are at the heart of the convergence theory. Metric regularity of mappings and prox-regularity of functions will be used extensively to study convergence rates of splitting algorithms; and (3) exploit some applications in signal processing and financial mathematics.******Seeing the success of my HQPs brings me joy. Amazingly, while postdoctoral fellows and graduate students can concentrate on theory developments, undergraduates can do some numerical experiments on algorithms and computations. I believe that this research will significantly advance our knowledge about applied variational analysis in theory, algorithms, and applications. People optimize. These much needed research results will have both local influence and global impact on the optimization community.

我的研究是在应用分析中。变分分析是凸分析和经典分析的扩展，以涵盖各种非不同函数和映射。凭借其开发的工具，在不同领域寻求应用程序，实际问题和计算算法至关重要。 ******设定值映射的一个重要结果是attouch-thera二元性，假设原始人具有解决方案。如果原始人没有解决方案，则尚不清楚是否存在Fenchel-Rockafellar二重性的类似于最大单调映射。 Epi-Convergence对于研究扩展实现功能的收敛至关重要。在此拓扑中，我们对非convex函数的最小化行为有什么看法？凸函数的近端平均值是现代凸分析中的一个强大工具，但其对非凸功能的概括却少得多。到目前为止，几乎所有分裂算法都需要两个操作员都是单调的。如果至少一个操作员不是单调的会发生什么？ Poliquin，Rockafellar，Thibault，Mordukhovich，Mordukhovich，Ioffe，Lewis等人系统地研究了功能和映射的度量规则的临时性，但是它们的算法后果等待着一项全面的研究。从凸函数到非凸函数，单调映射到非单调映射，这些概括在本质上很难，通常需要新的方法论。尽管已经取得了一些进展，但根本不令人满意。 （2）开发算法和局部收敛理论，用于求解非单调映射总和的零，并最大程度地减少非凸函数的总和。将检查Chen和Rockafellar，Bauschke，Combettes，Noll和Thera的作品。广义的局部非专用映射是融合理论的核心。映射的度量规则性和功能的较定期性将广泛用于研究分裂算法的收敛速率； （3）在信号处理和财务数学中利用一些应用程序。******看到我的HQP的成功使我感到高兴。令人惊讶的是，尽管博士后研究员和研究生可以专注于理论发展，但本科生可以在算法和计算上进行一些数字实验。我认为，这项研究将大大提高我们对理论，算法和应用中应用变异分析的知识。人们优化。这些急需的研究结果将对优化社区产生当地影响和全球影响。