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Modified Bfgs Method For Symmetric Nonlinear Equations

Posted on:2005-04-02Degree:MasterType:Thesis
Country:ChinaCandidate:G L YuanFull Text:PDF
GTID:2190360122498428Subject:Applied Mathematics
Abstract/Summary:PDF Full Text Request
The BFGS method is a well-known quasi-Newton method for solving unconstained optimization problems. The method only use the objective function and it's first-order derivative value without forming Hessian matix. Moreover, the method possesses the advantage which is quickly convengent speed and good numerical result. Powell and Werner(1978) proved the BFGS method is convergent for uniform convex functions respectively. Ritter(1981) proved the convergence of Bronden class. The study of many line search technique improved quasi-Newton method: Dennis and More(1974), Ghewank and Toint(1982) proved the superlinear convergence of BFGS method with step size one respectively. Wolfe(1969), (1971), Stoer(1975), Powell(1976), Warth and Werner(1977) gave the search technique accordingly. Nocedal(1989) gave new analysis tool for BFGS method. Rescent years, many authors proposed some modified BFGS method, such as, Fukushima, lqQi, zxWei, dhLi, etc., and proved the convergence of the modified methods.The Gauss-Newton method is well-known method for solving least-square problems firstly. Many authors made a great progress in studying of the method, Womersley(1985) proved the local quadratic convergence of Gauss-Newton methods under the assumption of strong uniqueness. Burke and Hem(1986) gave a Gauss-Newton approch to solving inequalities. Burke and Ferris(1993) studied a Gauss-Newton method for convex composite optimization. Li and Fukushima(1999) proposed a Gauss-Newton-based BFGS method for symmetric nonlinear equations. They proved the convergence and reported the numerical results.In this paper, we propose an approximate Gauss-Newton-based method for symmetric nonlinear equations which based on the study of Li and Fukushima and make a further study. We modify the update formula and get some better properties. The new method can ensure the positive property of update matrix. We give a modified algorithm and establish it's global and superlinear convergence under suitable conditions. Preliminary numerical results are reported.The paper is organized as follows:Chapter 1. Quasi-Newton EquationChapter 2. Gauss-Newton MethodChapter 3. A BFGS Method For Symmetric Nonlinear EquationsChapter 4. A Modified BFGS Method For Symmetric Nonlinear EquationsChapter 5. The Further Study of The Modified Method...
Keywords/Search Tags:BFGS method, Gauss-Newton method, Symmetric equations, Global convergence, Superlinear convergence
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