Quasi-multiplier rules for quasidifferentiable extremum problems

The purpose of the present article is to contribute to clarify the role of the Lagrange multipliers within the theory of the first order necessary optimality conditions for nonsmooth constrained optimization, when the directional derivatives of functions involved in the extremum problems are not sublinear. This task is accomplished in the particular case of quasidifferentiable problems with side constraints. In such setting, making use of the image-space approach, it is possible to establish a generalized (nonlinear) separation result by means of which a new Lagrange principle is obtained. According to this principle, which seems to fit better quasidifferentiable extremum problems than the classic one, the concept of linear multiplier is to be replaced with that of quasi-multiplier, a sublinear and continuous functional whose existence can be guaranteed under mild assumptions, even when classic multipliers fail to exist. Such as extension allows to formulate in terms of Lagrange function the known optimality necessary condition for unconstrained quasidifferentiable optimization expressed in form of quasidifferential inclusion. Along with this, other multiplier rules are established.