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Generalized reduced gradient and global Newton methods. (English) Zbl 0599.90102
Optimization and related fields, Proc. G. Stampacchia Int. Sch. Math., Erice/Italy 1984, Lect. Notes Math. 1190, 1-20 (1986).
[For the entire collection see Zbl 0581.00011.]
It is shown how to solve a system of n nonlinear equations $$F(x)=0$$, $$F: R^ n\to R^ n$$, by the global Newton method. In general the global Newton method uses the system of equations $$f(x)-\lambda (f(x^ 0)=0$$ where $$x^ 0$$ is some starting point such that $$f(x^ 0)\neq 0$$. An initial solution is $$\lambda =1$$, $$x=x^ 0$$. The method then tries forcing $$\lambda$$ to 0, and consequently x to some solution to $$f(x)=0.$$
To achieve this the following nonlinear programming problem is solved by use of the general reduced gradient method: min $$\lambda$$, subject to $$f(x)-\lambda (x^ 0)=0$$ and $$\lambda >0$$. The method is also applied to constrained optimization problems of the form: min $$f_ 0(x)$$, subject to $$f_ i(x)\leq 0$$, $$i=1,...,m$$, through use of the Kuhn-Tucker conditions. Numerical examples are given.
Reviewer: M.A.Hanson

MSC:
 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming 65H10 Numerical computation of solutions to systems of equations 65K05 Numerical mathematical programming methods
Zbl 0581.00011