The Nonmonotone Projected L-M Method For Convex Constrained Nonlinear Equations

To consider convergence of constrained nonlinear equations, many papers assume that the Hessian matrix is nonsingular. In this paper, we presented the nonmonotone projected Levenberg-Marquardt method for solving convex constrained nonliear equations. Under the local error bound near the solution, we prove that the new algorithm is globally and locally quardratic convergent.Firstly, we find a search direction d_k with Levenberg-Marquardt method. The projection will be used to maintain the feasibility of the iterate points. When the sufficiently descent conditiono is not satisfied, the inexact line search will be applied. Hence, if the sequence {x_k} generated by this method does not terminate finitely, then every limit point of the sequence is a stationary point of the problem.Moreover, under the local error bound condition, the sequence {x_k} is convergent at a quadratic rate.Finally, the numerical tests show that this new method is very effective.