On difference convexity of locally Lipschitz functions

Bacak, Miroslav; Borwein, Jonathan M.

doi:10.1080/02331931003770411

Cited by 49 publications

(48 citation statements)

References 22 publications

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“…DC functions are studied in many areas, such as optimization, game theory, variational analysis, spectral theory, and operator theory; see [1] for more details.…”

mentioning

confidence: 99%

A framework of discrete DC programming by discrete convex analysis

Maehara

Murota

2014

Math. Program.

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A theoretical framework of difference of discrete convex functions (discrete DC functions) and optimization problems for discrete DC functions is established. Standard results in continuous DC theory are exported to the discrete DC theory with the use of discrete convex analysis. A discrete DC algorithm, which is a discrete analogue of the continuous DC algorithm (Concave-Convex procedure in machine learning) is proposed. The algorithm contains the submodular-supermodular procedure as a special case. Exploiting the polyhedral structure of discrete convex functions, the algorithms tailored to specific types of discrete DC functions are proposed. Mathematics Subject Classification 90C27 · 90C46

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“…DC functions are studied in many areas, such as optimization, game theory, variational analysis, spectral theory, and operator theory; see [1] for more details.…”

mentioning

confidence: 99%

A framework of discrete DC programming by discrete convex analysis

Maehara

Murota

2014

Math. Program.

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“…The functional (1) is the difference of two continuous, convex functionals F and G so that we can build on existing results of DC (difference of convex functions, delta-convex functions) computation. For a recent overview on DC functions see [3].…”

Section: General Problemmentioning

confidence: 99%

Dithering by Differences of Convex Functions

Teuber¹,

Steidl²,

Gwosdek³

et al. 2011

SIAM J. Imaging Sci.

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Motivated by a recent halftoning method which is based on electrostatic principles, we analyze a halftoning framework where one minimizes a functional consisting of the difference of two convex functions (DC). One of them describes attracting forces caused by the image gray values, the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: the points are pairwise distinct, lie within the image frame and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional like its coercivity and propose to compute a minimizer by a forwardbackward splitting algorithm. We suggest to compute the special sums occurring in each iteration step by a fast summation technique based on the fast Fourier transform at non-equispaced knots which requires only O(m log m) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our dithering method.

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“…Indeed, optimization problems involving DC functions appear frequently in engineering, economics and other sciences. The reader is referred to the surveys [1,13,25] for relevant results, examples and applications of DC functions.…”

Section: Relation With DC Functions and Mapsmentioning

confidence: 99%

“…The following corollary was inspired by the absoluteness of difference convexity (see [1,Theorem 2.9]). …”

Section: Let Us Mention That Any Slice H Which Does Not Meet [D]mentioning

confidence: 99%

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Maps with the Radon–Nikodým Property

García-Lirola

Raja

2017

Set-Valued Var. Anal

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Abstract. We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikodým property to a "less linear" frame. We note that a certain part of the theory can be developed in rather great generality. Indeed, we establish that the elements of the dual which are "strongly slicing" for a given uniformly continuous dentable function form a dense G δ subset of the dual. As a consequence, the space of uniformly continuous dentable maps from a closed convex bounded set to a Banach space is a Banach space. However some interesting applications, as Stegall's variational principle, are no longer true beyond the usual hypotheses, sending us back to the classical case. Moreover, we study the connection between dentability and approximation by delta-convex functions for uniformly continuous functions. Finally, we show that the dentability of a set is closely related with the dentability of delta-convex maps defined on it.

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On difference convexity of locally Lipschitz functions

Cited by 49 publications

References 22 publications

A framework of discrete DC programming by discrete convex analysis

A framework of discrete DC programming by discrete convex analysis

Dithering by Differences of Convex Functions

Maps with the Radon–Nikodým Property

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