J Glob Optim (2010) 46:603–614 DOI 10.1007/s10898-009-9445-8 Some applications of the image space analysis to the duality theory for constrained extremum problems G. Mastroeni Received: 9 May 2009 / Accepted: 12 May 2009 / Published online: 24 May 2009 © Springer Science+Business Media, LLC. 2009 Abstract By means of the Image Space Analysis, duality properties of a constrained extremum problem are investigated. The analysis of the lower semicontinuity of the pertur- bation function, related to a right-hand side perturbation of the given problem, leads to a characterization of zero duality gap in the image space. Keywords Image space · Separation · Perturbation function 1 Introduction The Image Space Analysis (for short, ISA) [7] has shown to be a unifying scheme for studying constrained extremum problems, variational inequalities, and, more generally, can be applied to any kind of problem, say it ( P ), that can be expressed under the form of the impossibility of a parametric system. In this approach, the impossibility of such a system is reduced to the disjunction of two suitable subsets K and H of the Image Space (for short, IS) associated with ( P ). K is defined by the image of the functions involved in ( P ), while H is a convex cone that depends only on the class of problems to which ( P ) belongs. The disjunction of K and H can be proved by showing that they lie in two disjoint level sets of a suitable separating functional. In the case where ( P ) is a constrained extremum problem, several theoretical aspects can be developed, as duality, existence of optimal solutions, Lagrangian-type optimality condi- tions, penalty methods and regularity [6–9, 13, 16]. The subclass of separating functionals that fulfil the condition that the intersection of their positive level sets coincides with H, is said to be regular. Duality arises from the existence of a regular separating functional such that K is included in its non positive level set: this is shown to be equivalent to a saddle point condition for a generalized Lagrangian function associated with ( P )[7]. G. Mastroeni (B ) Department of Mathematics, University of Pisa, Pisa, Italy e-mail: mastroen@dm.unipi.it 123