Fixed-point quasi-Newtonian methods. (English) Zbl 0758.65043

Martínez, José Mario

SIAM J. Numer. Anal. 29, No. 5, 1413-1434 (1992).

This paper studies iterative methods of the form (1) \(x_{k+1}=\Phi(x_ k,E_ k)\) where \(x_ k\in\mathbb{R}^ n\), and \(E_ k\) belongs to some parameter space. Three examples of methods which may be written in this form are given, namely quasi-Newton methods for nonlinear systems, sequential quadratic programming and nonlinear complementarity.
Firstly, the author finds sufficient conditions for the sequence generated by (1) to be convergent to a fixed point of \(\Phi\) at a linear rate. Then, sufficient conditions for local convergence of the sequence at “ideal” rates are proved (the “ideal” iteration \(x_{k+1}=\Phi(x_ k,E_ *)\) has an ideal convergence rate \(r_ *\) which may be found).
The author then discusses convergence theory for least-change fixed-point iterations of quasi-Newton methods and ends by proving local and superlinear convergence results for a particular splitting of \(F(x)\) for the system \(F(x)=0\), of the form \(F(x)=F_ 1(x)+F_ 2(x)\) where, for fixed \(x\) and some \(B\), the nonlinear system \(F_ 1(z)+F_ 2(x)+B(z- x)=0\) is easy to solve for \(z\).

Reviewer: A.Swift (Palmerston North)

MSC:

65H10	Numerical computation of solutions to systems of equations
65K05	Numerical mathematical programming methods
90C20	Quadratic programming
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Keywords:

iterative methods; quasi-Newton methods; nonlinear systems; sequential quadratic programming; nonlinear complementarity; local convergence; least-change fixed-point iterations; superlinear convergence