Fenchel-Lagrange c-conjugate duality in infinite convex optimization

Jose Vidal, Dr. María Dolores Fajardo Gómez

In this talk we obtain a Fenchel-Lagrange dual problem for an infinite dimensional optimization primal one, via perturbational approach and applying a conjugation scheme called c-conjugation.

Using this approach, Fenchel-Lagrange dual problem is shown to be a combination of Fenchel and Lagrange dual problems obtained in former works. Motivated by this relation, the purpose of this talk is twofold. First, we will analyse the main inequalities that these three optimization problems satisfy, as well as sufficient conditions for equality among their optimal values. Secondly, we will develop two closedness-type regularity conditions via epigraphs, and a characterization for strong Fenchel-Lagrange duality in terms of the infimal convolution of two functions. As it happens in the classical context with the lower semicontinuity and convexity of the involved functions in the primal problem, the evenly convexity and properness will be a compulsory requirement since the conjugation scheme that we are applying is appropriate for the class of evenly convex functions, which can be characterized as those functions whose epigraph can be expressed as the intersection of an arbitrary family of open half-spaces.

Finally, we extend such conditions to the study of stable strong Fenchel-Lagrange duality.