An introduction to the theory of nonsmooth optimization

Abstract

This is an introductory overview of nonsmooth analysis, i.e. differential study of functions which are not Frechet differentiable everywhere.

In Section 1 we study how the Frechet derivative can be extended to functions which are only locally Lipschitzian. A convenient generalization is due to F, H. Clarke; we define it, give some of its properties and compare it to some other proposals. In Section 2 we study various subclasses of locally Lipschitzian functions, including semi smooth and max functions. Section 3 is exclusively devoted to convex functions, with a special study of the approximate subdifferential, illustrated by some examples (indicators, conjugacy. max functions, quadratic and piece wise linear functions).

The paper deals only with that part from nonsmooth analysis that has possible applications in nonsmooth optimization. Furthermore the style is definitely intuitive and geometric.

Keywords:

• Nonsmooth analysis
• nonsmooth optimization
• nondifferentiable analysis
• nondifferentiable optimization
• convex analysis

AMS 1980 Subject Classification:

• Primary:90 C 30
• Primary:26 B 25
• Secondary:26 B 05
• Secondary:90 C 25