5 Lipschitz Nonsmooth Optimization
Marius Durea, Radu Strugariu · 2014
Lipschitz Nonsmooth OptimizationSmooth and convex functions, which we discussed in the previous two chapters, are also (locally) Lipschitz.This chapter aims to introduce a more general framework, which provides optimality conditions for problems with Lipschitz data.We first present the theory of generalized gradients for locally Lipschitz functions, developed by Frank H. Clarke in the 1970's, and we then pass to a more general framework, by giving some comprehensive elements of the modern theory developed by Boris S. Mordukhovich and his collaborators in the last three decades.Of course, we limit our exposition to the finite dimensional spaces framework, but both of these theories can be developed in much more general situations.It is beyond the scope of this book to discuss the case of multifunctions (i.e., mappings with values given by sets), and of the associated generalized differentiation objects (derivatives and coderivatives, associated by means of corresponding tangent and normal cones), although many interesting results, which link the whole machinery of calculus for functions, multifunctions and tangent cones exist in literature.