The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

Abstract

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.

Keywords:

• Global convergence
• Inconsistent
• Least l ₂-norm solution
• Levenberg–Marquardt method
• Local quadratic convergence
• Nonlinear inequalities

AMS Subject Classification:

• 90C30
• 65K05

ACKNOWLEDGMENTS

The first author's work is supported by the National Natural Science Foundation of China 10671203, 70621001, and 70531040; the second author's work is supported by the National Natural Science Foundation of China 10571134 and the Natural Science Foundation of Tianjin 07JCYBJC05200; and the third author's work is supported by the Hong Kong Research Grant Council.