A superlinearly convergent projection algorithm for solving the convex inequality problem

Abstract

The Convex Inequality Problem (CIP), i.e., find x∈ R n such that Ax=b, g(x)⩽0, where A is an p× n matrix, b∈ R m and g(.) : R n→ R m is a convex function, has been solved by projection algorithms possessing a linear rate of convergence. We propose a projection algorithm that exhibits global and superlinear rate of convergence under reasonable assumptions. Convergence is ensured if the CIP is not empty. A direction of search is found by solving a quadratic programming problem (the projection step). As opposed to previous algorithms no special stepsize procedure is necessary to ensure a superlinear rate of convergence. We suggest a possible application of this algorithm for solving convex constrained Linear Complementarity Problems, i.e., find x∈ R n such that x⩾0, Ax+b⩾0, x,Ax+b =0, g(x)⩽0. A is an n× n positive semidefinite matrix and g(.) : R n→ R m is a convex function.