Global convergence and finite termination of a class of smooth penalty function algorithms

Abstract

In this paper, based on a new class of asymptotic l ₁ exact penalty functions, we propose a smooth penalty function for solving nonlinear programming problems. One of the main features of our algorithm is that at each iteration, we do not need to solve the global minimum of penalty functions. Furthermore, global convergence property is established without requiring any constraint qualifications. By addressing perturbation functions, we obtain that the lower semi-continuity of the perturbation function at zero is a necessary and sufficient condition to ensure the convergence of the objective function values generated by the algorithm to the optimal value of the primal problem. Since the perturbation function is only dependent on the data of the primal problem, these results enable us to check the convergence property of the algorithm in advance. Finally, we discuss the finite termination of the algorithm when the solution set is non-degenerate or weakly sharp.

Keywords:

• smooth penalty function
• perturbation function
• global convergence
• finite termination
• weak sharpness

AMS Subject Classifications: :

• 90C30
• 49M37
• 65K05

Acknowledgements

We are indebted to the anonymous referees for their valuable suggestions and remarks, which essentially improved the presentation of this paper. The research of this paper was supported by the National Natural Science Foundation of P.R.China (10971118, 11101248, 10901096) and the Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2009AL019).