Computation and Theory for a Class of Nonsmooth Functions

这个项目将研究一类非光滑方程的理论和计算方面，这些方程涉及被称为法线映射的函数。这些法线映射表示了优化问题或平衡问题解的必要条件，因此，求解一个涉及这样一个函数的方程就相当于满足了优化问题或平衡问题解的必要条件。许多实际问题都涉及到这些问题的解决，因此，拟议的工作如果成功，将具有很大的实际实用价值。一般说来，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。在理论方面，将研究这类函数的几个全局拓扑方面。特别是，将研究法线映射是同胚的特殊条件，以便更一般地研究其临界值的性质，并探索将全局同伦方法从包含光滑函数的方程扩展到涉及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着眼于产生能够帮助解决困难的非线性优化或平衡问题的方法，对于这些问题来说，没有好的起始值。

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.