Global convergence of a class of trust region algorithms for optimization with simple bounds. (English) Zbl 0643.65031

Conn, A. R.; Gould, N. I. M.; Toint, Ph. L.

SIAM J. Numer. Anal. 25, No. 2, 433-460 (1988).

This paper extends the known global convergence properties of trust region algorithms for un-constrained optimization to the case where bounds on the variables are presented in the form f(x)\(=^{!}Min\), \(\ell_ i\leq x_ i\leq u_ i\), \((i=1,...,n)\). These extensions are obtained by generalizing the classical notion of Cauchy point and by considering generaion see Zbl 0641.00024.]
For the problem: \(\epsilon y''+p(x)y'=f(x)\), \(-\alpha y(0)+y'(0)=\alpha_ 0\), \(y(1)=\alpha_ 1\), p(x)\(\geq \bar p>0\), the cubic spline collocation method is derived. Uniform convergence of first order on locally bounded mesh is achieved. The method has second order convergence for fixed \(\epsilon\).

MSC:

65K05	Numerical mathematical programming methods
90C30	Nonlinear programming

Keywords:

global convergence; trust region algorithms; constrained optimization; Cauchy point; cubic spline collocation method; Uniform convergence

Citations:

Zbl 0641.00024