Global Convergence of a Trust Region Algorithm for Nonlinear Inequality Constrained Optimization Problems

Abstract

In the paper, a new trust region algorithm is given for nonlinear inequality constrained optimization problems. Motivated by a dual problem introduced by Han and Mangasarian [Han, S. P., Mangasarian, O. L. (1983). A dual differentiable exact penalty function. Math. Programming 25:293–306], which is a nonnegatively constrained maximization problem, we construct a trust region algorithm for solving the dual problem. At each iteration, we only need to minimize a quadratic subproblem with simple bound constraints. Under the condition that the iterate sequence generated by the algorithm is contained in some bounded closed set, any accumulation point of the sequence is a Karush–Kuhn–Tucker point of the original problem.

Key Words:

• Nonlinear optimization
• Trust region method
• Inequality constraints
• Global convergence

Mathematical Subject Classification:

• 65K05
• 49D37

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable comments, which lead to the improvements of the presentation of the paper. The first author's work is supported by NSFC 10001031, 10171095 and the Hi-Tech Research and Development Program of China (863 Program) 2002AA103061 and the second author's work is supported by NSFC 10271002.