Abstract Convexity of Functions with Respectto the Set of Lipschitz (Concave) Functions

Gorokhovik, Valentin V.; Tykoun, A. S.

doi:10.1134/s0081543820040057

Cited by 6 publications

(12 citation statements)

References 10 publications

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“…In this section, for the convenience of readers, we recall basic definitions of abstract convexity and regularly abstract convexity, for more details we refer to the monographs [5][6][7] and to the papers [15,16]. Throughout this section unless other specified X is an abstract set, that is not equipped with any topological, algebraic or other structures; elements of X will be called points.…”

Section: Preliminaries On H-convex and Regularly H-convex Functionsmentioning

confidence: 99%

“…Following [15,16] we call a function f : X Þ Ñ R regularly H-convex if S ḿax pH, f q ‰ ∅ and f pxq " supthpxq | h P S ḿax pH, f qu for all x P X.…”

Section: Preliminaries On H-convex and Regularly H-convex Functionsmentioning

confidence: 99%

“…Like the classical convex analysis the study of abstract convexity is mainly stimulated by numerous applications in optimization. In particular, in the framework of abstract convexity criteria for global minima and maxima [5,[14][15][16] as well as the duality results [3,[17][18][19][20][21] were derived.…”

Section: Introductionmentioning

confidence: 99%

“…The special class of abstract H-convex functions called regularly abstract H-convex ones was introduced in [15,16].…”

Section: Introductionmentioning

confidence: 99%

“…From Section 3 and later on we study the particular class of regularly H-convex functions, when H is the set L p CpX, Rq of real-valued Lipschitz continuous concave (in classical sence) functions defined on a real normed space X. It was proved in [15,16] that an extended-real-valued function f C-convex as it is bounded from below by an affine continuous function. Thus the class of L p C-convex functions is a nontrivial extension of the class of lower semicontinuous convex functions.…”

Section: Introductionmentioning

confidence: 99%

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Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Gorokhovik¹

2022

Preprint

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Given a set H of functions defined on a set X, à function f : X Þ Ñ R is called abstract H-convex if it is the upper envelope of its H-minorants, i.e., such its minorants which belong to the set H; and f is called regularly abstract H-convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) H-minorants. In the paper we first present the basic notions of (regular) H-convexity for the case when H is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a H-convex function to be regularly H-convex is formulated.The goal of the paper is to study the particular class of regularly H-convex functions, when H is the set L p CpX, Rq of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function f : X Þ Ñ R to be L p C-convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each L p C-convex function is regularly L p C-convex as well. We focus on L p Csubdifferentiability of functions at a given point. We prove that the set of points at which an L p C-convex function is L p C-subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset L p C θ of the set L p C consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of L p C θ -subgradient and L p C θ -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for L p C θ -subdifferentials as well as L p C θ -subdifferential conditions for global extremum points are established. Symmetric notions of abstract L q C-concavity and L q C-superdifferentiability of functions where L q C :" L q CpX, Rq is the set of Lipschitz continuous convex functions are also considered.

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Section: Preliminaries On H-convex and Regularly H-convex Functionsmentioning

confidence: 99%

“…Following [15,16] we call a function f : X Þ Ñ R regularly H-convex if S ḿax pH, f q ‰ ∅ and f pxq " supthpxq | h P S ḿax pH, f qu for all x P X.…”

Section: Preliminaries On H-convex and Regularly H-convex Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The special class of abstract H-convex functions called regularly abstract H-convex ones was introduced in [15,16].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Gorokhovik¹

2022

Preprint

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Zero duality gap conditions via abstract convexity

et al. 2021

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On duality for nonconvex minimization problems within the framework of abstract convexity

Bednarczuk

Syga

2021

Optimization

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We study conjugate and Lagrange dualities for composite optimization problems within the framework of abstract convexity. We provide conditions for zero duality gap in conjugate duality. For Lagrange duality, intersection property is applied to obtain zero duality gap. Connection between Lagrange dual and conjugate dual is also established. Examples related to convex and weakly convex functions are given.

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Abstract Convexity of Functions with Respectto the Set of Lipschitz (Concave) Functions

Cited by 6 publications

References 10 publications

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Zero duality gap conditions via abstract convexity

On duality for nonconvex minimization problems within the framework of abstract convexity

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