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.

我的研究方向是应用变分分析。变分分析是凸分析和经典分析的扩展，涵盖了各种非微分函数和映射。随着其工具的完善，寻求其在不同领域的应用、实际问题和计算算法是必不可少的。******集值映射的一个重要结果是Attouch-Thera对偶性，它假设原函数有一个解。如果原初无解，则不清楚是否存在极大单调映射的fenchell - rockafellar对偶的类似物。π收敛性是研究扩展实值函数收敛性的必要条件。在这种拓扑结构中，关于非凸函数的最小化行为我们能说些什么呢？凸函数的近平均是现代凸分析中的一个有力工具，但其推广到非凸函数的研究却很少。到目前为止，几乎所有的分割算法都要求两个算子都是单调的。如果至少有一个算子不是单调的，会发生什么？Poliquin、Rockafellar、Thibault、Mordukhovich、Ioffe、Lewis等人已经系统地研究了函数的亲正则性和映射的度量正则性，但它们的算法结果有待全面的研究。从凸函数到非凸函数，从单调映射到非单调映射，这些推广本质上是困难的，往往需要新的方法。虽然取得了一些进展，但一点也不令人满意。******在本提案中，我计划(1)研究Attouch-Thera的对偶性，包络的非凸函数的一般最小化性质，近端平均应用和扩展，以及Mordukhovich的协导数分析和Rockafellar的原导数分析包络的二阶非光滑分析；(2)发展求解非单调映射和的零的算法和局部收敛理论，以及最小化非凸函数和。陈和洛克菲勒、鲍施克、孔贝特斯、诺尔和西拉的作品将被展出。广义局部非扩张映射是收敛理论的核心。映射的度量正则性和函数的准正则性将广泛用于研究分裂算法的收敛速度；(3)在信号处理和金融数学方面的应用。******看到我的hqp的成功给我带来了快乐。令人惊讶的是，当博士后和研究生可以专注于理论发展时，本科生可以做一些关于算法和计算的数值实验。我相信这项研究将大大提高我们对应用变分分析在理论、算法和应用方面的认识。人们优化。这些迫切需要的研究成果将对优化界产生局部影响和全球影响。