EXACT BARRIER FUNCTION METHODS FOR LIPSCHITZ PROGRAMS

The purpose of this paper is twofold. First we consider a class of non differentiable penalty functions for constrained Lipschitz programs an d then we show how these penalty functions can be employed to solve a constrained Lipschitz program. The penalty functions considered incorp orate a barrier term which makes their value go to infinity on the bou ndary of a perturbation of the feasible set. Exploiting this fact it i s possible to prove, under mild compactness and regularity assumptions , a complete correspondence between the unconstrained minimization of the penalty functions and the solution of the constrained program, thu s showing that the penalty functions are exact according to the defini tion introduced in [17]. Motivated by these results, we propose some a lgorithm models and study their convergence properties. We show that, even when the assumptions used to establish the exactness of the penal ty functions are not satisfied, every limit point of the sequence prod uced by, a basic algorithm model is an extended stationary point accor ding to the definition given in [8]. Then, based on this analysis and on the one previously carried out on the penalty functions, we study t he consequence on the convergence properties of increasingly demanding assumptions. In particular we show that under the same assumptions us ed to establish the exactness properties of the penalty functions, it is possible to guarantee that a limit point at least exists, and that any such limit point is a KKT point for the constrained problem.