Application of a Satisfactory Approximation of an Admissible Set for Solving Optimization Problems

This work deals with the properties and construction principles of a satisfactory approximation of a set of admissible solutions for a conditional optimization problem. The replacement of an initial admissible set by its satisfactory approximation allows one to construct finite algorithms for the methods of internal and external points (methods of penalty functions or methods of centers) with the stopping criterion which ensures the required accuracy of the solution. Necessary and sufficient conditions for producing external and internal satisfactory approximations of an admissible set are proved. One of the feasible ways for setting a satisfactory approximation of an admissible set is formulated. This way can be used for the development of algorithms that ensure the required accuracy in a finite number of iterations.