Newton-type methods for nonsmooth equations with nonisolated solutions

最优化和变分问题的必要条件往往导致互补系统。在其计算处理中，这些系统经常被重新表述为非光滑方程。由于解的非孤立性是广义纳什均衡问题或拟变分不等式等几类问题的典型属性，具有非孤立解的互补系统越来越受到人们的关注。然后，互补系统的重新表述导致具有退化和非孤立解的方程。最近，牛顿型方法被发展出来，可以处理这种困难的情况。这些方法是基于将互补系统重新表示为分段光滑方程。到目前为止，对分段光滑方程的限制强烈地限制了这些方法全球化的可能性。因此，该项目首先旨在开发新的牛顿型方法，该方法可以处理互补系统的某些其他重构。即使解是退化的和非孤立的，新方法在弱条件下也将表现出局部超线性收敛。通过这种方式，可以访问现有的成熟的全球化概念。此外，作为推广，该项目旨在开发一种新的算法范式来求解具有非孤立解的非光滑方程。这包括用于计算迭代序列的子问题的设计，局部收敛理论，以及对超越经典互补系统的问题的适用性

Necessary conditions for optimization and variational problems often lead to complementarity systems. For its computational treatment these systems are frequently reformulated as nonsmooth equations. Complementarity systems with nonisolated solutions become of increasing interest since nonisolatedness of solutions is a typical attribute of several problems classes like generalized Nash equilibrium problems or quasi-variational inequalities. The reformulation of complementarity systems then leads to equations with degenerate and nonisolated solutions. Recently, Newton-type methods have been developed that can deal with such difficult situations. These methods are based on the reformulation of complementarity systems as piecewise smooth equations. By now, the restriction to piecewise smooth equations strongly limits the possibilities of globalizing these methods. Therefore, the project firstly aims at developing new Newton-type methods that can deal with certain other reformulations of complementarity systems. The new methods shall exhibit local superlinear convergence under weak conditions even if solutions are degenerate and nonisolated. In this way, existing mature concepts of globalization can be accessed. Moreover, as a generalization, the project aims at developing a new algorithmic paradigm for solving nonsmooth equations with nonisolated solutions. This includes the design of subproblems for computing the iteration sequence, local convergence theory, and the applicability to problems that go beyond classical complementarity systems