Powell, M. J. D. Variable metric methods for constrained optimization. (English) Zbl 0536.90076 Mathematical programming, 11th int. Symp., Bonn 1982, 288-311 (1983). Summary: [For the entire collection see Zbl 0533.00035.] Variable metric methods solve nonlinearly constrained optimization problems, using calculated first derivatives and a single positive definite matrix, which holds second derivative information that is obtained automatically. The theory of these methods is shown by analysing the global and local convergence properties of a basic algorithm, and we find that superlinear convergence requires less second derivative information than in the unconstrained case. Moreover, in order to avoid the difficulties of inconsistent linear approximations to constraints, careful consideration is given to the calculation of search directions by unconstrained minimization subproblems. The Maratos effect and relations to reduced gradient algorithms are studied briefly. Cited in 22 Documents MSC: 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming 65K05 Numerical mathematical programming methods Keywords:variable metric methods; nonlinearly constrained optimization problems; global and local convergence; superlinear convergence; calculation of search directions; Maratos effect; reduced gradient algorithms Citations:Zbl 0533.00035 PDFBibTeX XML