The conditions of minimum for a smooth function on the boundary of a quasidifferntiable set

In this paper, we consider problems of mathematical programming with nonsmooth constraints of equality type given by quasidifferentiable functions. By using the technique of upper convex approximations, developed by B. N. Pshenichy, necessary conditions of extremum for such problems are established. The Lagrange multipliers signs are specified by virtue of the fact that one can construct whole familers of upper convex approximations for quasidifferentiable function and thus the minimum points in such extremal problems are characterized more precisely. Also the simplest problem of calculus of variations with free right hand side is considered, where the left end of the trajectory starts on the boundary of the convex set. The transversality condition at the left end of the trajectory is improved provided sertain sufficient conditons hold.