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Solving Constrained Optimization Problems And Symmetric Variational Inequality Kkt Systems Bfgs Method

Posted on:2004-09-16Degree:MasterType:Thesis
Country:ChinaCandidate:J W ZhangFull Text:PDF
GTID:2190360092490580Subject:Basic mathematics
Abstract/Summary:PDF Full Text Request
In this thesis, we consider the BFGS method for solving the constrained optimizatiom problem and the symmetric variational inequality problem. First, we reformulate the KKT systems of the constrained optimization problem and the symmetric variational inequality prpoblem into equivalent nonsrnooth equations by using the so called NCP functions. Based on these reformulations, we propose BFGS methods for solving these nonsmooth equations.For the nonsmooth equation reformulation of the KKT system of the constrained optimization problem, we develop the method on the basis of a generalized Newton method. We use the matrices generated by a modified BFGS formula to replace the generalized Jacobian matrices in the generalized Newton method. By virtue of a non-monotone line search, we propose a Gauss-Newton-based BFGS method for solving the nonsmooth euqation. Under mild conditions, we prove the global and superlinear convergence of the method. Moreover, we prove that after finite iterations, the unit step is always accepted and the proposed method essentially reduces to the modified BGS method with the unit steplength.We also propose a BFGS method for solving the nonsmooth equation reformulation of the KKT system of the symmetric variational inequality problem. We introduce a parameter in the non-monotone line search. By changing the parameter in a suitable way, we get a descent direction of the BFGS method. We then develop a BFGS method for solving the nonsmooth equation. The method possess some descent property. Under mild conditions, we establish the global and superlinear convergence of the proposed method. We also show that the unit steplength is essentially accepted.
Keywords/Search Tags:constrained optimization problems, symmetric variational inequality prob-lems, KKT system, BFGS method, global convergence, superlinear convergene, general-ized derivative, semismooth, strongly semismooth.
PDF Full Text Request
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