Stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization

This paper introduces an algorithm for minimizing a single-variable locally Lipschitz function subject to a like function being nonpositive. The method combines polyhedral and quadratic approximation, a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point. The convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases. The algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms. Research sponsored, in part, by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-83-0210. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.