The results presented in this book originate from the last decade
research work of the author in the ?eld of duality theory in convex
optimization. The reputation of duality in the optimization theory comes
mainly from the major role that it plays in formulating necessary and
suf?cient optimality conditions and, consequently, in
generatingdifferent algorithmic approachesfor solving mathematical
programming problems. The investigations made in this work prove the
importance of the duality theory beyond these aspects and emphasize its
strong connections with different topics in convex analysis, nonlinear
analysis, functional analysis and in the theory of monotone operators.
The ?rst part of the book brings to the attention of the reader the
perturbation approach as a fundamental tool for developing the so-called
conjugate duality t- ory. The classical Lagrange and Fenchel duality
approaches are particular instances of this general concept. More than
that, the generalized interior point regularity conditions stated in the
past for the two mentioned situations turn out to be p- ticularizations
of the ones given in this general setting. In our investigations, the
perturbationapproachrepresentsthestartingpointforderivingnewdualityconcepts
for several classes of convex optimization problems. Moreover, via this
approach, generalized Moreau-Rockafellar formulae are provided and, in
connection with them, a new class of regularity conditions, called
closedness-type conditions, for both stable strong duality and strong
duality is introduced. By stable strong duality we understand the
situation in which strong duality still holds whenever perturbing the
objective function of the primal problem with a linear continuous
functional.
