Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme

Fajardo, Maria Dolores; Vidal-Núñez, José

doi:10.1007/s10957-017-1209-x

Cited by 12 publications

(12 citation statements)

References 18 publications

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“…Now, the idea is to prove that K cannot be closed, and to do that, we shall check that given any (x * , y * , α, γ) ∈ K, the point (x * , 0, 0, γ) ∈ cl K\K. It is clear that (x * , 0, 0, γ) / ∈ K because this point does not belong to any epigraph of the c -elementary functions that build epi h. By (9), K is the solution set of the system…”

Section: Properties Of E -Convex Setsmentioning

confidence: 99%

“…[5] states the analysis of regularity conditions for Lagrange duality. In [9] Fenchel-Lagrange duality is considered, where dual problems are expressed via the c-conjugates of the functions involved in the primal problem. [7] studies sufficient conditions and characterizations for stable strong duality in this generalized framework for Fenchel and Lagrange dualities.…”

Section: Introductionmentioning

confidence: 99%

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E′-Convex Sets and Functions: Properties and Characterizations

Fajardo

Vidal-Núñez

2020

Vietnam J. Math.

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The main properties of evenly convex sets and functions have been deeply studied by different authors, and a duality theory for evenly convex optimization problems has been well developed as well. In this theory, the notion of e-convexity appears as a necessary requirement for obtaining important results in strong and stable strong duality. This fact has motivated the authors to study possible properties of this kind of convexity in sets and functions, which is closely connected to even convexity.

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Section: Properties Of E -Convex Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

E′-Convex Sets and Functions: Properties and Characterizations

Fajardo

Vidal-Núñez

2020

Vietnam J. Math.

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“…with these the duality theory plays a fundamental role in the analysis of optimization and variational problems. The reader can refer to [1], [8], [9], [25], [27] and their references for more details on this topic. It not only provides a powerful theoretical tool in the analysis of these problems, but also paves the way to designing new algorithms for solving them.…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

“…Consequently, the key to our success is the formulation of the Lemmas 4.4,4.5 and Propositions 3.2,4.1, 4.2, 4.6, without which it is hardly ever possible to establish any duality to the problem with second order DSIs and DFIs. To the best of our knowledge, there are a few papers (see [9], [16], [17], [25] and references therein) devoted to duality problems of first order DFIs. Building on these results, we then treat dual results according to the dual operations of addition and infimal convolution of convex functions [1], [9], [11], [14], [15].…”

Section: Elimhan N Mahmudovmentioning

confidence: 99%

Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions

Mahmudov¹

2021

Evolution Equations &Amp; Control Theory

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The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. At the end of the paper duality in problems with second order linear discrete and continuous models and model of control problem with polyhedral DFIs are considered.

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“…Take any point y ∈ Y and denote a = Ψ((0, ·), (0, ·), ·) c ′ (y) ∈ R. Then, according to (9) it yields that (y, a) ∈ Pr Y ×R (epi Ψ c ′ ). Hence, by (10), (y, a) ∈ epi inf x∈X Ψ c ′ (x, ·) and (12) holds.…”

Section: Converse Duality and Biconjugationmentioning

confidence: 99%

New duality results for evenly convex optimization problems

2020

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We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and csubdifferentials are given. Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality also by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.MSC Subject Classification. 52A20, 26B25, 90C25, 49N15.

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Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme

Cited by 12 publications

References 18 publications

E′-Convex Sets and Functions: Properties and Characterizations

E′-Convex Sets and Functions: Properties and Characterizations

Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions

New duality results for evenly convex optimization problems

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