Exact Barrier Function Methods for Lipschitz Programs

The purposeof this paper is twofold. First we considera class of nondiierentiablepenaltyfunctions for constrained Lipschitz programs and then we show how these penalty functions can be employed to actually solve a constrained Lipschitz program. The penalty functions considered incorporate a barrier term which makes their value go to innnity on the boundary of a perturbation of the feasible set. Exploiting this fact it is possible to prove, under mild compactness and regularity assumptions, a complete correspondence between the unconstrained minimization of the penalty functions and the solutions of the constrained program, thus showing that the penalty functions are exact according to the deenition introduced in 17]. Motivated by these results, we then propose some algorithm models and study their convergence properties. We show that, even when the assumptions used to establish the exactness of the penalty functions are not satissed, every limit point of the sequence produced by a basic algorithm model is an extended stationary point according to the deenitiongiven in 8].Then, based on this analysis and on the one previously carried out on the penalty function, we study the consequences on the convergence properties of increasingly demanding assumptions. In particular we show that under the same assumptions used 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.