Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization

Abstract

In this paper, a new quadratic smoothing approximation to the $l_1$ exact penalty function is proposed. It is shown that under certain conditions, if there exists a global minimizer of the original constrained optimization problem in the ''interior'' of the feasible set of the original constrained optimization problem, then any global minimizer of the smoothed penalty problem is a global minimizer of the original constrained optimization problem when the penalty parameter is sufficiently large; and if the feasible region of the original constrained optimization problem is ''robust'', then any global minimizer of the smoothed penalty problem is a feasible approximate global minimizer of the original constrained optimization problem when the penalty parameter is sufficiently large, and the precision of the approximation can be set in advance. Some numerical examples are given to illustrate that constrained optimization problems can be well solved by the present smoothing scheme.