Applied Variational Analysis: Theory, Algorithms, and Applications

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，涵盖了各种非可微函数和映射。随着其工具的成熟，寻求它们在不同领域、实际问题和计算算法中的应用是必不可少的。*集值映射的一个重要结果是Atouch-Thera对偶，它假设原映射有解。如果原函数没有解，则不清楚是否存在与极大单调映象的Fichel-Rockafella对偶类似的解。EPI-收敛是研究广义实值函数收敛的基础。在这种拓扑中，关于非凸函数的极小化行为，我们能说些什么呢？凸函数的近似平均是现代凸函数分析中的一个强有力的工具，但其推广到非凸函数的研究较少。到目前为止，几乎所有的分裂算法都要求两个算子都是单调的。如果至少有一个运算符不是单调的，会发生什么？Poliquin，Rockafella，Thibault，Mordukhovich，Ioffe，Lewis等人已经系统地研究了函数的Prox正则性和映射的度量正则性，但它们的算法结果还有待全面的研究。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，它们往往需要新的方法。虽然已经取得了一些进展，但并不令人满意。*在这个建议中，我计划(1)研究Atouch-Thera的对偶性，非凸函数的包络、近平均应用和扩张的一般极小化性质，以及利用Mordukhovich的余导数分析和Rockafella的原导数分析对包络的二阶非光滑分析；(2)发展求解非单调映射和的零点以及最小化非凸函数和的算法和局部收敛理论。陈冠希和罗卡菲勒、鲍施克、康贝茨、诺尔和特拉的作品将被审查。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正则性将被广泛地用于研究分裂算法的收敛速度；以及(3)开发一些在信号处理和金融数学中的应用。*看到我的HQP的成功，我得到了喜悦。令人惊讶的是，博士后研究员和研究生可以专注于理论发展，而本科生可以做一些关于算法和计算的数值实验。我相信，这项研究将极大地促进我们在理论、算法和应用方面对应用变分分析的了解。人们会进行优化。这些亟需的研究成果将对优化社区产生局部和全球影响。

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.