A New Smooth Method for the l1 Exact Penalty Function for Inequality Constrained Optimization

Abstract

Exact penalty function methods for the solution of constrained optimization problem are based on the construction of a function whose unconstrained minimizing points are also solution of the constrained problem. One of the popular exact penalty functions is l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> exact penalty function. However l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> exact penalty function is not a smooth function. In this paper, we propose a new method for smoothing the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem problem and of the original optimization problem. We develop an efficient algorithm for solving the optimization problem based the smoothed penalty function and prove the convergence of the algorithm.