A Smoothing Projected Levenberg-Marquardt Type Algorithm for Solving Constrained Equations

Abstract

In this article, we consider the problem of finding a solution of a nonsmooth constrained (and not necessarily square) system of equations. We first reformulate the original problem as an equivalent system of equations with nonnegative constraints, and then present a smoothing projected Levenberg-Marquardt type algorithm to solve the reformulated system, which solves a strictly convex quadratic program at each iteration. We show that this algorithm not only converges globally, but also converges locally superlinearly under an error bound assumption that is much weaker than the standard nonsingularity condition. Some numerical results for the presented algorithm indicate that the algorithm works quite well in practice.

Keywords:

• Convergence
• Nonsmooth constrained equations
• Projected Levenberg-Marquardt algorithm
• Semi-smoothness
• Smoothing technique

2000 Mathematics Subject Classification:

• 65K05
• 65H10
• 90C30
• 90C33
• 90C53