On strong and total Lagrange duality for convex optimization problems

Abstract

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.

Characterizations of ɛ-duality gap statements for constrained optimization problems

“…[3][4][5]) in the direction of ε-vertical closedness can be considered, too. Note that one can formulate also ε-optimality conditions statements for (P) and its considered dual problems, extending thus the corresponding optimality conditions statements from [6,7]. Moreover, ε-Farkas type results can be given for the considered problem by combining the statements from this paper with ideas from [10].…”

“…Motivated by recent results on stable strong and total duality for constrained convex optimization problems in [2,6,7,9,13,17] and the ones on zero duality gap in [15,16] we introduce in this paper several regularity conditions which characterize ε-duality gap statements (with ε ≥ 0) for a constrained optimization problem and its Lagrange and FenchelLagrange dual problems, respectively. The regularity conditions we provide in Section 2 are based on epigraphs, while the ones in Section 3 on ε-subdifferentials.…”

“…Motivated by the characterizations of the stable strong duality from [6,7] we begin this section with several equivalent representations of various instances of ε-duality gap for (P) and its duals by means of epigraphs.…”

“…Taking ε = 0, Theorem 3.1 becomes [6,Theorem 5] without the topological and convexity assumptions on the involved functions, Corollary 3.2 is [6, Theorem 6], while Corollary 3.3 turns into [7,Theorem 3]. Note also that by considering (RCL) for ( ) = 0 whenever ∈ X one can extend (analogously to Theorem 3.12, see also Remarks 3.13 and 3.14) [6, Theorem 9] towards ε-duality gap and the same can be done for their counterparts corresponding to the other considered duals.…”

Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces

“…The Moreau-Rockafellar formula is somewhat similar to the condition GBCQ 1 (f, A), which was introduced in [5] to study the Lagrange and FenchelLagrange dualities for conical programming. Clearly, in the case when A = id X , g = δ A , and the condition GBCQ 1 (0, A) is satisfied, the Moreau-Rockafellar formula (5.1) is equivalent to the condition GBCQ 1 (f, A) in [5] (see [4,5] for details).…”