Separation and regularity in the image space

Abstract

The optimality for a constrained extremum problem can be studied proving the disjunction between suitable subsets of the image space associated to the problem. This disjunction can be stated showing that these sets are linear or conically separated [1,2,3]. Under the “image regularity condition” (in short IRC) in [2], it has been proved the existence of generalized Karush—Kuhn—Tucker multipliers associated to a conic separation. Moreover it has been proved that IRC is equivalent to the calmness introduced by Clarke and the existence of an exact penalty function. Here we study the possibility of relating the local image regularity condition with a generic nonlinear separation not necessarily conic or linear and we provide refinements of some results obtained in [2].