Convergence criteria for usable direction methods

Abstract

The nonlinear programming problem P: maximize f(x) subject to g(x) nonnegative, is considered where f and g are continuously differentiable functions defined on Euclidean n-space, and g is vector-valued. Convergence theorems are obtained that possess the following characteristics: they (1) identify properties of usable direction methods which ensure finite limit points of sequences generated by them are Kuhn-Tucker points of problem P; (2) present convergence criteria that are readily applicable to many usable direction methods; and (3) suggest ways to construct usable direction methods which generate sequences whose finite limit points are Kuhn-Tucker points of problem P. The problem of constructing and analyzing steplength functions which meet the criteria of the convergence theorems is addressed from a unified viewpoint. In addition, several classes of usable direction algorithms are presented whose convergence properties are analyzed via the convergence theorems.