Convex Analysis and Monotone Operators: Forward and Backward

单调算子在数学及其应用的许多领域中发挥着核心作用，例如在最优化和微分方程式理论中。凸分析与单调算子理论之间的许多联系早在50年前就已为人所知，但在2002年之后，人们对它们进行了深入的研究，当时人们重新发现了所谓的Fitzpatrick函数及其推广，即代表函数。这个项目的主要范围是通过凸分析的新进展和概念，例如一般正则性条件，提供关于单调算子(组合)的新陈述，例如关于最大单调性的陈述。该项目分为五个目标。首先，计划对代表性函数和次微分概念进行研究。然后，我们打算将关于单调算子的几个结果从自反推广到一般的Banach空间。然后我们将讨论单调算子的合成和扩张。在第四个目标的框架内，计划对所谓的对角次微分算子进行研究。由于许多有趣的问题(如极小化问题、互补问题、变分不等式问题)都可以归结为单调包含问题，我们还计划提供寻找单调算子组合零点的分离型算法，并在具体应用中实现它们。

The monotone operators play a central role in many areas of Mathematics and its applications, for instance in Optimization and in the theory of Differential Equations. Many connections between Convex Analysis and the theory of Monotone Operators are known for around 50 years, however they were intensively studied after 2002, when the so-called Fitzpatrick function and its generalizations known as the representative functions were rediscovered. The main scope of this project is to deliver new statements on (combinations of) monotone operators, e.g. concerning maximal monotonicity, by means of the new advances and concepts from Convex Analysis, for instance general regularity conditions. The project is structured into five objectives. Firstly, investigations on representative functions and subdifferential concepts are planned. Afterwards, we intend to extend several results on monotone operators from reflexive to general Banach spaces. Then we will deal with compositions and extensions of monotone operators. Within the framework of the fourth objective investigations on so-called diagonal subdifferential operators are scheduled. Because many interesting problems (e.g. minimization problems, complementarity problems, variational inequalities) can be cast as monotone inclusions, we also plan to deliver splitting type algorithms for finding zeros of combinations of monotone operators and to implement them on concrete applications.