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Semismooth Equations, Newton's Class Method

Posted on:2008-06-01Degree:MasterType:Thesis
Country:ChinaCandidate:N LuoFull Text:PDF
GTID:2190360215499992Subject:Applied Mathematics
Abstract/Summary:PDF Full Text Request
Optimization, which makes research on how to find the optimal solution among many fea-sible projects, is widely applied in many fields such as finance, trade, management and scientificresearch.How to solve the system of nonlinear equations is an important component of optimization.A classical algorithm for the problem is Newton's method. The method is attractive because itconverges rapidly for any sufficiently good initial point x0. However, solving a system of linearequations (the Newton equations) exactly at each stage can be expensive if the number of un-knowns is large and may not be justified when xk is far from a solution. But the drawbacks can beovercome to some extent if we solved linear equations inexactly and using affine transformationwill improve the result and the feasibility. So, we propose an affine inexact Newton method.For solving bound-constrained semi-smooth equations, a Newton-like method with projec-tion is proposed. In order to generate feasible iterates, we introduce Newton-like method that isaugmented by the projection onto feasible set. It is proved that the additional projection does notaffect the local super-linear convergence speed.In order to obtain global convergence, we describe an algorithm called affine scaling interiormethod. The basic idea is to require a step back-tracking along the Newton-like step by the strictinterior feasibility and line search technique, because line search is an important technique whichcan obtain global convergence. The frame of the paper is following:In chapter 2, an affine inexact Newton method for the system of nonlinear equations is pro-posed. we will find it has local super-linear convergence rate under some reasonable conditions.In chapter 3, We proposes affine inexact Newton method with projection and affine scaling interiormethod for solving bound-constrained semi-smooth equations. The affine scaling interior methodis a combination of inexact Newton method and line search method. If the iterate direction doesn'tsatisfy acceptable rules, we can get new step which can decrease the function value by using bothline search and interior point backtracking technique. We will give a full proof of the global andlocal super-linear convergence results. Furthermore, numerical results are given to indicate thatthe algorithm is useful and effective in practice. Finally, in chapter 5, a summary of this paper ismade and research direction is proposed.
Keywords/Search Tags:nonlinear equations, semi-smooth, bound constrained, inexact Newton methods, affine scaling, projection, interior point, local convergence rate, global convergence
PDF Full Text Request
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