"Variational analysis of monotone operators and beyond: theory, algorithms and applications"

我的研究将把变分分析应用于单调算子。变分分析是一种 包含各种非可微函数(凸的或非凸的)和映射的凸分析的扩展。变分分析有更强大的工具，如余导数、原导数、原正则性、变分原理和许多现代技术。单调算子在最优化、凸分析、不动点理论和偏微分方程中扮演着重要的角色。到目前为止，非光滑分析的二阶理论并不令人满意；为了取得良好的进展，有必要对单调算子的余导数和图导数进行系统的分析。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。如果一个运算符不是单调的，会发生什么？即使是交替预测，在这个方向上也没有取得什么进展。虽然单调算符场为变分分析提供了一个很好的试验场，但要超越单调性(或凸性)，就必须依赖于变分分析中的现代技术。本文的目的是利用变分(非凸)分析方法研究单调算子，发展求解其中一个算子是单调而另一个不一定是单调的包含问题的局部收敛理论和算法，并在图像处理和数理经济中开拓一些应用。文中还将讨论Simons的q-正集和SSDB空间。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，这一研究将极大地促进我们对单调(n-单调)算子的分析性质和结构的了解，获得对SSDB空间的新见解，并将分裂算法带到一个新的水平。实际影响是求解单调算子与Lipschitz映射之和的零点的有效算法。然后，这些可以应用于工程和癌症研究中的建模，以及经济均衡。我的研究的成功将使加拿大受益，并产生全球影响。

My research will apply variational analysis to monotone operators. Variational analysis is an extension of convex analysis that encompasses a variety of nondifferential functions (convex or nonconvex) and mappings. Variational analysis has more powerful tools such as coderivative, proto-derivative, prox-regularity, variational principle and many modern techniques. Monotone operators play key roles in Optimization, Convex Analysis, Fixed Point Theory, and Partial Differential Equations. Up to now, the second order theory of nonsmooth analysis is not satisfactory; to make good progress, it is worthwhile to systematically analyze coderivatives and graphical derivatives of monotone operators. Up to now, almost all splitting algorithms need both operators to be monotone. What happens if one operator is not monotone? Even for alternating projections, little progress has been made in this direction. While monotone operator field provides an excellent test ground for variational analysis, to go beyond monotonicity (or convexity) one has to rely on modern techniques in variational analysis. The goal of this proposal is to study monotone operators by variational (nonconvex) analysis, develop local convergence theory and algorithms for solving inclusions in which one operator is monotone and the other not necessarily monotone, and exploit some applications in image processing and mathematical economics. The q-positive sets and SSDB spaces by Simons will also be explored. Amazingly, while post-doctoral 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 the analytical properties and structures of monotone (n-monotone) operators, gain new insights on SSDB spaces and bring splitting algorithms to a new level. Practical impacts are efficient algorithms for solving for zeros of the sum of a monotone operator and a Lipschitz mapping. These could then be applied to modelling in engineering and cancer research, and economic equilibrium. The success of my research will benefit Canada and have global impacts.