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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

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