Computation and Theory for a Class of Nonsmooth Functions

本项目将研究一类涉及称为法线映射的函数的非光滑方程的理论和计算方面。这些法线映射表示了求解最优问题或平衡问题的必要条件，因此，求解包含此类函数的方程等于满足求解最优问题或平衡问题的必要条件。许多实际问题都涉及这些问题的解决，因此，拟议的工作如果成功，将具有实质性的实际效用。一般来说，非光滑法线映射具有相当多的结构，许多一阶局部可微函数理论已经推广到这些映射中。在理论方面，将研究这些函数的几个全局拓扑方面。特别地，我们将研究法线映射为同胚的特殊条件，以便更广泛地研究它们的临界值的性质，并探索将全局同伦方法从包含光滑函数的方程扩展到包含法线映射的方程的可能性。在计算方面，将采用一些嵌入和同伦方法来研究它们的数值行为。这些计算研究将特别针对产生有助于解决难的非线性优化或平衡问题的方法，这些问题没有良好的起始值。

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.