On the Lagrange Multiplier Rule for Minimizing Sequences

Abstract

In the paper, an optimization problem with equality-type constraints is studied. It is assumed that the minimizing function and the functions defining the constraints are Frechet differentiable, the set of the admissible points is nonempty and the minimizing function is bounded below on the set of admissible points. Under these assumptions we obtain an estimate of the derivative of the Lagrange function. Moreover, we prove the existence of a minimizing sequence \(\{x^n\}\) and a sequence of unit Lagrange multipliers \(\{\lambda^n\}\) such that the sequence of the values of derivative of the Lagrange function at the point \((x^n, \lambda^n)\) tends zero. This result is a generalization of the known assertion stating that for a bounded below differentiable function \(f\) there exists a minimizing sequence \(\{x^n\}\) such that the values of the derivative \(f'(x^n)\) tend to zero. As an auxiliary tool, there was introduced and studied the property of the directional covering for mappings between normed spaces. There were obtained sufficient conditions of directional covering for Frechet differentiable mappings.