Necessary Conditions for an Extremum, Penalty Functions and Regularity

Abstract

The aim of this paper is to show that the three concepts mentioned in the title are closely related when considering the most general optimization problems. Each of these concepts has been studied in numerous works where different relations among them have been established. Here we give the most general statements and we show that in the presence of regularity the necessary conditions for an extremum in the form of the Lagrange multipliers rule are always fulfilled. It should be emphasized that for the most complete description of the point of extremum in problems with non-smooth constraints the optimality conditions must be formulated not in the form of a single rule of multipliers but in the form of the whole family of such rules. Only this makes it possible to avoid a situation when necessary conditions are fulfilled at points which trivially cannot be optimal.