A comparison of alternative c-conjugate dual problems in infinite convex optimization

Abstract

In this work we obtain a Fenchel-Lagrange dual problem for an innite dimensional optimization primal one, via perturbational approach and using a conjugation scheme called c-conjugation instead of classical Fenchel conjugation. This scheme is based on the generalized convex conjugation theory. We analyze some inequalities between the optimal values of Fenchel, Lagrange and Fenchel-Lagrange dual problems and we establish sucient conditions under which they are equal. Examples where such inequalities are strictly fullled are provided. Finally, we study the relations between the optimal solutions and the solvability of the three mentioned dual problems.

New duality results for evenly convex optimization problems

“…By cl f we denote the lower semicontinuous hull of f , which is the function whose epigraph equals cl(epi f ). A function f is lower semicontinuous, lsc in brief, if for all x ∈ X, f (x) = cl f (x), and e-convex if epi f is e-convex in X × R. Clearly, any lsc convex function is e-convex, but the converse does not hold in general as one can see in [8,Ex. 2.1].…”

“…Taking any point (x, y, Ψ c ′ (x, y)) ∈ epi Ψ c ′ , we have (y, Ψ c ′ (x, y)) ∈ C and, for all λ ≥ 0, (y, Ψ c ′ (x, y)) + λ(0, 1) = (y, Ψ c ′ (x, y) + λ) ∈ C. According to Lemma 2.1, (0, 1) ∈ rec C, and since C is functionally representable by assumption, due to Remark 2.1 we obtain C = epi h with h(x) = inf {a ∈ R : (x, a) ∈ C} . If we show that h = e-conv p, we have, using also (8), the chain epi Ψ((0, ·), (0, ·), ·) c ′ = epi(e-conv p) = epi h = C = e-conv(Pr Y ×R (epi Ψ c ′ )).…”

“…In [ 18 ] we analysed the problem of stable strong duality, and deduced Fenchel and Lagrange type duality statements for unconstrained and constrained optimization problems, respectively. In [ 19 ] the Fenchel–Lagrange dual problem of a (primal) minimization problem, whose involved functions do not need to be e-convex a priori, was derived. Furthermore, some relations between the optimal values of the Fenchel, Lagrange and Fenchel–Lagrange dual problems were presented.…”

“…Furthermore, some relations between the optimal values of the Fenchel, Lagrange and Fenchel–Lagrange dual problems were presented. Finally, in [ 20 ] we used the formulation of the Fenchel–Lagrange dual problem from [ 19 ] to derive a characterization of strong Fenchel–Lagrange duality.…”

E′-Convex Sets and Functions: Properties and Characterizations

“…[7] studies sufficient conditions and characterizations for stable strong duality in this generalized framework for Fenchel and Lagrange dualities. In addition, a comparison of the optimal values and solutions of the three alternative dual problems (Fenchel, Lagrange and Fenchel-Lagrange) is achieved in [8]. Finally, in the recent paper [4], converse and total duality as well as new results on subdifferential theory for e-convex functions have been also studied, developed initially in [15] .…”