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

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 \(z\).

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) |