Globalization techniques for solving nonlinear problems and applications

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Globalization techniques for solving nonlinear problems and applications

 

Author: Chen, Jinhai
Title: Globalization techniques for solving nonlinear problems and applications
Degree: Ph.D.
Year: 2009
Subject: Hong Kong Polytechnic University -- Dissertations.
Nonsmooth optimization.
Department: Dept. of Applied Mathematics
Pages: xii, 140 p. : ill. ; 30 cm.
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b2321026
URI: http://theses.lib.polyu.edu.hk/handle/200/4849
Abstract: This thesis is concentrated on the study of global techniques to the numerical solution of nonsmooth problems. These problems are directly relevant to the variational inequality problems, complementarity problems, heat transmission problems in medium, parabolic obstacle problems within financial mathematics, and control-state constrained optimal control problems. These global approaches are discussed deeply on convergence theories and computational results. A vital point in the implementation of the global approaches to the minimization of a nonsmooth (merit) function, such as Newton methods based on path search, line search, and trust region algorithms, is the calculation of a (generalized) Jacobian matrix equation effectively, especially for large-scale problems. In Chapter 2, we consider a Krylov subspace strategy for the underlying global methods to solving nonsmooth equations. Such strategy has a main advantage of computing a generalized Jacobian matrix equation, especially, in many applications where the (generalized) Jacobian matrix is not practically computable, or is expensive to obtain. Another global strategy is to consider avoiding the minimization of a nonsmooth (merit) function, pseudotransient continuation in Chapter 3 may be a nice choice for this purpose. Many practical problems have certain structures; if we could find these structures, we may design more suitable algorithms. These algorithms have not only global convergence, but also have special properties, for example: finite termination, monotonicity. Indeed, the Newton-type methods considered to solve piece-wise systems in Chapter 4 have a remarkable monotone convergence. These piece-wise systems arise from the discretizations of heat transmission problems in a medium, parabolic obstacle problems in financial mathematics. Other strategies, for example: smoothing strategy, nonmonotone strategy, etc., have also rather good effects in most of the practical implementations, even if in the infinite dimensional spaces. To show this, numerical solution of optimal control problems subject to mixed control-state constraints has been investigated in Chapter 5. The necessary conditions of the optimal control problems are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function, the local minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Ba-nach spaces. This nonlinear and nonsmooth equation is solved by inexact nonsmooth and smoothing Newton methods. The globalized methods are developed in a very general setting that allows for non-monotonicity of squared residual norm values at subsequent iterates. Numerical examples are presented to demonstrate the efficiency of these presented approaches.

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