Gradient methods in the variational problem with free ends

The efficiency of techniques of nonsmooth analysis and the theory of exact penalty functions for solving various problems of the calculus of variations was demonstrated in [1]. In this paper, some results of application of the approach described in [1] to the problem with free ends are presented. A “new” form of necessary conditions is obtained, and, based on them, new numerical algorithms (direct methods) of steepest descent and conjugate directions are constructed. The “natural boundary conditions” are derived in an elementary way from them.