Theorems of the alternative for multifunctions with applications to optimization: General results

Theorems of the alternative and separation theorems have been shown to be very useful concepts in constrained extremum problems (see, for instance, Refs. 1–12). Their use has stressed the concept of image of a constrained extremum problem, which has turned out to be a powerful and promising tool for investigating the main aspects of optimization (see Refs. 13 and 19). It should be pointed out that, in this approach, a finite-dimensional image problem can be associated to the given extremum problem, even if this is infinite-dimensional and provided that its constraints are expressed by functionals. Such a development can be carried on by means of theorems of the alternative for systems of single-valued functions.In this paper, theorems of the alternative for systems of multifunctions are studied, some general properties are stated, and connections with known results investigated. It is shown how the present approach can be used to analyze extremum problems, where the image of the domain of the constraining functions belongs to a functional space. Such a development will be carried on in a subsequent paper.Useful discussions with O. Ferrero and C. Zlinescu are gratefully acknowledged.