Existence of Exact Penalty and Its Stability for Nonconvex Constrained Optimization Problems in Banach Spaces

Abstract

In this paper we use the penalty approach in order to study two constrained minimization problems. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of cost functions, constraint functions and the right-hand side of constraints.

Existence and stability of exact penalty for optimization problems with mixed constraints

“…As we have already mentioned the main result of the present paper is a generalization of the results of [14][15][16][17]. In [14] we considered the problem (P) with one constraint function (m = 1) and established a very simple sufficient condition for the exact penalty property.…”

“…In the present paper we obtain a generalization of the results of [14][15][16][17]. We study the stability of the generalized exact penalty property under perturbations of the functions f 0 , f 1 , .…”

“…The notion of exact penalization was introduced by Eremin [7] and Zangwill [8] for use in the development of algorithms for nonlinear constrained optimization. Since that time exact penalty functions have continued to play a key role in the theory of mathematical programming [9][10][11][12][13][14][15][16][17]. For more discussions and various applications of exact penalization to various constrained optimization problems see [1,2,4,6].…”

“…In this paper we establish the existence of a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. This novel approach in the penalty type methods was used in [14][15][16][17].…”

“…In [14][15][16][17] and in this paper we are interested in approximate solutions of the unconstrained penalized problem and in approximate solutions of the corresponding constrained problem. Under certain assumptions which hold for a large class of problems we show the existence of a constant Λ 0 > 0 such that the following property holds:…”