Iteration Functions in Some Nonsmooth Optimization Algorithms

Recently, several globally convergent model algorithms based on iteration functions have been proposed for solving nonsmooth optimization problems. In particular, Pang, Han and Rangaraj proposed such an algorithm for minimizing a locally Lipschitzian function. We determine properties of iteration functions (calculus, existence); we also identify characteristics of functions that possess iteration functions. We show that a locally Lipschitzian function has a Pang-Han-Rangaraj iteration function only when the function is pseudo-regular (in the sense of Borwein), and that a subsmooth (lower-C¹) function always has a Pang-Han-Rangaraj iteration function.