An Effective Method for Studying Extremal Problems

Abstract

The homotopy method (or the parameter continuation method) first appeared in the middle of the 19th century and is being actively developed at the present time. In this paper, we consider a deformation (homotopy) method for studying extremal problems. Examples of this method are illustrated for well-known nonlinear analysis problems. One of the most general and efficient schemes for applying the homotopy method to the qualitative study of operator equations (with a completely continuous operator) was developed by Leray and Schauder. In this scheme, the parameter is included linearly. The article gives examples of applications of homotopy method to the study of extreme problems of variation calculus. The basic design is as follows: if during the deformation of the variational problem its extremal remains isolated and at any value of the deformation parameter this extremal achieves a minimum, then it implements the minimum of the variational problem under study for all values of the parameter.

Keywords

• homotopy method
• nonlinear programming problems
• variations calculus
• deformation principle
• stability of gradient systems