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

Reformulation semi-lisse appliquée au problème de complémentarité

Ce mémoire fait une revue des notions élémentaires concernant le problème de complé- mentarité. On y fait aussi un survol des principales méthodes connues pour le résoudre. Plus précisément, on s’intéresse à la méthode de Newton semi-lisse. Un article proposant une légère modification à cette méthode est présenté. Cette nouvelle méthode compétitive est démontrée convergente. Un second article traitant de la complexité itérative de la méthode de Harker et Pang est aussi introduit

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

Solving dynamic contact problems with local refinement in space and time

International audienceFrictional dynamic contact problems with complex geometries are a challenging task from the compu tational as well as from the analytical point of view since they generally involve space and time multi scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local struc ture and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix --- The full report.

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) 0 ≤ x ⊥ (Mx+q) ≥ 0 can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x,Mx+q)=0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.L'algorithme Newton-min, utilisé pour résoudre le problème de complémentarité linéaire (PCL) 0 ≤ x ⊥ (Mx+q) ≥ 0 peut être interprété comme un algorithme de Newton non lisse sans globalisation cherchant à résoudre le système d'équations linéaires par morceaux min(x,Mx+q)=0, qui est équivalent au PCL. Lorsque M est une M-matrice d'ordre n, on sait que l'algorithme converge en au plus n itérations. Nous montrons dans cet article que ce résultat ne tient plus lorsque M est une P-matrice d'ordre n ≥ 3 ; l'algorithme peut en effet cycler dans ce cas. On a toutefois la convergence de l'algorithme pour une P-matrice d'ordre 1 ou 2