Hybrid Newton-type method for a class of semismooth equations

In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite-difference Newton method to an inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We report also several numerical results obtained applying the method to suitable reformulations of well-known nonlinear complementarity problem

A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization

We propose a novel trust region method for solving a class of nonsmooth and nonconvex composite-type optimization problems. The approach embeds inexact semismooth Newton steps for finding zeros of a normal map-based stationarity measure for the problem in a trust region framework. Based on a new merit function and acceptance mechanism, global convergence and transition to fast local q-superlinear convergence are established under standard conditions. In addition, we verify that the proposed trust region globalization is compatible with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence results. We further derive new normal map-based representations of the associated second-order optimality conditions that have direct connections to the local assumptions required for fast convergence. Finally, we study the behavior of our algorithm when the Hessian matrix of the smooth part of the objective function is approximated by BFGS updates. We successfully link the KL theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type condition to show superlinear convergence of the quasi-Newton version of our method. Numerical experiments on sparse logistic regression and image compression illustrate the efficiency of the proposed algorithm.Comment: 56 page

On a Nonsmooth Gauss–Newton Algorithms for Solving Nonlinear Complementarity Problems

In this paper, we propose a new version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems based on the transformation to the nonsmooth equation, which is equivalent to some unconstrained optimization problem. The B-differential plays the role of the derivative. We present two types of algorithms (usual and inexact), which have superlinear and global convergence for semismooth cases. These results can be applied to efficiently find all solutions of the nonlinear complementarity problems under some mild assumptions. The results of the numerical tests are attached as a complement of the theoretical considerations

On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given