A smoothing self-adaptive Levenberg–Marquardt algorithm for solving system of nonlinear inequalities

Abstract

In this paper, we consider the smoothing self-adaptive Levenberg–Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H( x) = 0. A smoothing self-adaptive Levenberg–Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg–Marquardt parameter μ k is chosen as the product of μ k = ∥ H k ∥ δ with δ ∈ (0, 2] being a positive constant. We will show that if ∥ H k ∥ δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈ (0, 1), while quadratically for δ ∈ [1, 2]. Numerical results show that the new method performs very well for system of inequalities.