Computation and Theory for a Class of Nonsmooth Functions

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This project will investigate theoretical and computational aspects of a class of nonsmooth equations involving functions called normal maps. These normal maps express the necessary conditions for solution of optimization or equilibrium problems, so solving an equation involving such a function amounts to satisfying the necessary conditions for solution of optimization or for equilibrium. A great many practical problems involve the solution of such problems, so the proposed work, if successful, would have substantial practical utility. Generally nonsmooth normal maps have a considerable amount of structure, and much of the first-order local theory of differentiable functions has already been extended to these maps. On the theoretical side, several global topological aspects of such functions will be investigated. In particular, special conditions will be studied under which normal maps will be homeomorphisms, in order to investigate more generally, the nature of their critical values, and to explore the possibility of extending global homotopy approaches from equations containing smooth functions to those involving normal maps. On the computation side, some of the embedding and homotopy methods will be implemented to investigate their numerical behavior. These computational investigations will be aimed particularly at producing methods that could help in solving hard nonlinear optimization or equilibrium problems for which good starting values are not available.