Newton-type methods for nonsmooth equations with nonisolated solutions

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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