The usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear. In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty. We apply this theory to solve the until now unsolved problem of finding, in the infinite dimensional case, the Lagrange multipliers associated to optimization problems or to variational inequalities. As an example, we find the Lagrange multiplier associated to a general elastic-plastic torsion problem.

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We consider a time-dependent economic market in order to show the existence of time-dependent market equilibrium (which we call dynamic equilibrium). The model we are concerned with is the spatial price equilibrium model in the presence of excesses of supplies and of demands.This kind of network problem is directly incorporated into the Variational Inequality model, which provides not only the existence, but also the computation, the stability and the sensitivity of the equilibrium patterns.The study of the time-dependent model seems to be important because it allows us to follow the evolution in time of prices and of commodities.

In this paper the authors present an infinite dimensional duality theory for optimization problems and evolutionary variational inequalities where the constraint sets are given by inequalities and equalities. The difficulties arising from the structure of the constraint set are overcome by means of generalized constraint qualification assumptions based on the concept of quasi relative interior of a convex set. An application to a general evolutionary network model, which includes as special cases traffic, spatial price and financial equilibrium problems, concludes the paper.