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

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Continuation method for nonlinear complementarity problems via normal maps

Cataloged from PDF version of article.In a recent paper by Chen and Mangasarian (C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications 2 (1996), 97±138) a class of parametric smoothing functions has been proposed to approximate the plus function present in many optimization and complementarity related problems. This paper uses these smoothing functions to approximate the normal map formulation of nonlinear complementarity problems (NCP). Properties of the smoothing function are investigated based on the density functions that de®nes the smooth approximations. A continuation method is then proposed to solve the NCPs arising from the approximations. Su cient conditions are provided to guarantee the boundedness of the solution trajectory. Furthermore, the structure of the subproblems arising in the proposed continuation method is analyzed for di erent choices of smoothing functions. Computational results of the continuation method are reported. Ó 1999 Elsevier Science B.V. All rights reserved

Implementation of a continuation method for nonlinear complementarity problems via normal maps

Ankara : Department of Industrial Engineering and Institute of Engineering and Sciences, Bilkent Univ., 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references.In this thesis, a continuation method for nonlinear complementarity problems via normal maps that is developed by Chen, Harker and Pinar [8] is implemented. This continuation method uses the smooth function to approximate the normal map reformulation of nonlinear complementarity problems. The algorithm is implemented and tested with two different plussmoothing functions namely interior point plus-smooth function and piecewise quadratic plus-smoothing function. These two functions are compared. Testing of the algorithm is made with several known problems.Erkan, AliM.S

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems

The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function

AbstractThe nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. By introducing a new smoothing NCP-function, the problem is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is proposed for solving the nonlinear complementarity problem with P0-function (P0-NCP) based on the new smoothing NCP-function. The proposed algorithm solves only one linear system of equations and performs only one line search per iteration. Without requiring strict complementarity assumption at the P0-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions