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起着核心作用，例如在最优化和微分方程理论中。凸分析和单调算子理论之间的许多联系已经有大约50年的历史了，但是在2002年之后，当所谓的菲茨帕特里克函数及其推广被称为代表函数时，它们被深入研究。该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。该项目分为五个目标。首先，计划对代表函数和次微分概念进行研究。然后，我们打算把自反单调算子的一些结果推广到一般的Banach空间。然后我们将处理单调算子的合成和扩张。在框架内的第四个目标调查所谓的对角次微分算子的时间表。由于许多有趣的问题（例如，最小化问题，互补问题，变分不等式）可以被转换为单调包含，我们还计划提供分裂型算法，寻找零的单调算子的组合，并实现它们的具体应用。

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.