Perturbations of extremal problems with constraints and necessary optimality conditions
The methods of perturbations occupy a significant place in the contemporary investigations in the theory of extremal problems, differential equations, differential games, in the construction of numerical methods, etc. The various perturbations can be conventionally divided into two groups. The first group contains the perturbations for which the initial problem is included in the family of problems of the same type. With their aid one has constructed various methods of solutions of nonlinear equations, of the duality theory of extremal problems, one has obtained sufficient conditions for an extremum, etc. An example is the derivation of sufficient conditions for a minimum in the classical calculus of variations, based on the construction of the fields of extremals [see \textit{G. A. Bliss}, ``Lectures on the calculus of variations'' (1947; Zbl 0036.344)], or the method of continuation with respect to the parameter [\textit{D. F. Davidenko}, Dokl. Akad. Nauk SSSR 88, 601-602 (1953; Zbl 0050.121)] for the solution of nonlinear equations. However, the perturbations of this group are situated at a certain distance from the content of the present paper. This survey is devoted to the first group of perturbations. In them the initial extremal problem is imbedded in a family of problems, approximating it in some sense, and the perturbed problems have to be qualitatively simpler than the initial one. The results for the initial problem (as a rule, these are necessary extremum conditions) are obtained by a limiting process with respect to the perturbation parameter, following the analysis and the transformations of the approximating problems. The developed perturbation methods allow us to remove the constraints for the extremal problems and to use the already existing technical arsenal. They may serve as a definite heuristic means of investigation.
scientific article; zbMATH DE number 4198906
scientific article