A New Unified Path to Smoothing Nonsmooth Exact Penalty Function for the Constrained Optimization

Bingzhuang Liu

doi:10.4236/ojop.2021.103005

Abstract

We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.

Highlights

We consider the following constrained optimization problem min f ( x)(P) s.t. g j ( x) ≤ 0, j = 1, 2, m, (1)where f, g j : Rn → R, j =1, 2, m are continuously differentiable functions
In the penalty function method, the original constraint conditions are reflected to the new objective function by constructing penalty function, and the original constrained optimization problem is transformed into a series of unconstrained optimization problems
Many scholars have proposed smooth approximations to the classical exact penalty functions, which can be found in the references ([2]-[14]), and different penalty algorithms have been given to solve different optimization problems

Summary

Introduction

Where f , g j : Rn → R, j =1, 2, , m are continuously differentiable functions. This model has important applications in many fields such as industry, engineering, and computational science. Many scholars have proposed smooth approximations to the classical exact penalty functions, which can be found in the references ([2]-[14]), and different penalty algorithms have been given to solve different optimization problems. In [10] [11] and [14], smooth approximations to l1 penalty function were proposed for nonlinear inequality constrained optimization problems. Different smoothing penalty functions were proposed in [13] to solve the global optimization problems. To solve the problem (P), [7] proposed two smooth approximations to the exact penalty function m. On the basis of the proposed smoothing penalty functions, a new approximate algorithm is established, and the convergence of the algorithm is discussed under appropriate conditions. The above assumption is common since if it is not satisfied, we can take the place of f0 ( x) by e f0 (x) +1

Approximately Smoothing Exact Penalty Functions

Smooth Penalty Algorithm and Its Convergence

Conclusions