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We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved. Numerical results show that the new algorithm is efficient. [ABSTRACT FROM AUTHOR]

Abstract: We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature. [Copyright &y& Elsevier]

Abstract: We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. [Copyright &y& Elsevier]

We develop an iterative method for solving the KKT system of the semi-infinite programming (SIP) problem. At each iteration, we solve the KKT system of a nonlinear programming problem with finite constraints by a semismooth Newton method. The algorithm either terminates at a KKT point of the SIP problem in finitely many iterations or generates an infinite sequence of iterates whose any accumulation point is a KKT point of the problem. We also analyse the convergence rate of the method. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper, we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations. At each iteration, a trust region subproblem is solved. The solution of the trust region subproblem provides a descent direction for the norm of a smoothing function. By using a backtracking line search, a steplength is determined. The proposed method shares advantages of trust region methods and line search methods. Under appropriate conditions, the proposed method is proved to be globally and superlinearly convergent. In particular, we show that after finitely many iterations, the unit step is always accepted and the method reduces to a smoothing Newton method. [Copyright &y& Elsevier]