Directionally Lipschitzian Functions and Subdifferential Calculus 论文

摘要

The theory of subgradients of convex functions is recognized for its many applications to optimization and differential equations (for example, Hamiltonian systems, monotone operators). F. H. Clarke has extended the theory to non-convex functions that are merely lower semicontinuous and used it to derive necessary conditions for non-smooth, non-convex problems in optimal control and mathematical programming. For locally Lipschitzian functions, he has proved a number of rules for subgradient calculation that generalize the ones previously known for convex functions. This paper extends such rules to non-convex functions that are not necessarily locally Lipschitzian. The two main operations considered are the addition of functions and the composition of a function with a differentiable mapping. The theorems are strong enough to cover the main results known in the convex case.