# zbMATH — the first resource for mathematics

A global convergence theory for the Celis-Dennis-Tapia trust-region algorithm for constrained optimization. (English) Zbl 0725.65061
For the equality constrained nonlinear programming problem, several approaches are discussed to prevent inconsistent constraints in the quadratic subproblem. One possible way was proposed by M. R. Celis, J. E. Dennis and R. A. Tapia [Numerical optimization. Proc. SIAM, Conf., Boulder/Colo. 1984, 71-82 (1985; Zbl 0566.65048)], to replace the set of linear constraints in the subproblem by one norm constraint subject to an adaptable tolerance. This algorithm is considered furtheron and modified by the author.
The $$L_ 2$$-augmented Lagrangian function is used as a merit function for evaluating the actual and predicted reduction. Under some standard assumption, the global convergence of the algorithm is shown, i.e. the convergence when starting from an arbitrary initial point. To achieve the result, a couple of auxiliary lemmata have to be proved, e.g. suitable bounds for the actual and predicted change of merit function and the boundedness of the penalty parameter.

##### MSC:
 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming
Full Text: