A New Augmented Lagrangian Objective Penalty Function for Constrained Optimization Problems

Ying Zheng,Zhiqing Meng

doi:10.4236/ojop.2017.62004

Abstract

In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.

Highlights

Augmented Lagrangian penalty functions are effective approaches to inequality constrained optimization
When the KKT condition holds for convex programming its saddle point exists
Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is proved under some conditions

Summary

Introduction

Augmented Lagrangian penalty functions are effective approaches to inequality constrained optimization. Kaya gave an augmented Lagrangian scheme for a general optimization problem, and established for this update primal-dual convergence the augmented penalty method in [8] When it comes to computation, to apply these methods, lots of Lagrangian parameters or penalty parameters need to be adjusted to solve some unconstrained optimization dual problems, which make it difficult to obtain an optimization solution to the original problem. Meng et al studied properties of dual and saddle points of the augmented Lagrangian objective penalty function in [16].

Augmented Lagrangian Objective Penalty Function

Conclusion