Uniformly convex functions on Banach spaces

Borwein, Jonathan M.; Guirao, Antonio José; Hájek, Petr; Vanderwerff, Jon D.

doi:10.1090/s0002-9939-08-09630-5

Cited by 39 publications

(36 citation statements)

References 13 publications

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“…Proposition 5.1 Let X be a uniformly smooth Banach space. Then, for any x = 0 and y, we have For instance, it is known [10] that if ρ(u) ≤ γ u q , 1 < q ≤ 2, then E( f, x, q) is a uniformly smooth convex function with modulus of smoothness of order u q . Next,…”

Section: Discussionmentioning

confidence: 99%

Greedy Approximation in Convex Optimization

Temlyakov

2015

Constr Approx

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We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of a few elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. The problem of approximation of a given element of a Banach space by linear combinations of elements from a given system (dictionary) is well studied in nonlinear approximation theory. At first glance, the settings of approximation and optimization problems are very different. In the approximation problem, an element is given and our task is to find a sparse approximation of it. In optimization theory, an energy function is given and we should find an approximate sparse solution to the minimization problem. It turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular the greedy approximation technique, can be adjusted for finding a sparse solution of an optimization problem.

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Section: Discussionmentioning

confidence: 99%

Greedy Approximation in Convex Optimization

Temlyakov

2015

Constr Approx

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“…The following theorem follows from their results (see, for instance, Theorem 2.3 in [3]), and in particular, for q = 2 see Theorem 3.4 in [8].…”

Section: Final Remarksmentioning

confidence: 75%

“…Hajek and J. Vanderwerff [3] studied the uniform convexity of functions on Banach spaces. The following theorem follows from their results (see, for instance, Theorem 2.3 in [3]), and in particular, for q = 2 see Theorem 3.4 in [8].…”

Section: Final Remarksmentioning

confidence: 99%

More on convexity and smoothness of operators

Cheng

2010

Journal of Mathematical Analysis and Applications

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Keywords: Convexity SmoothnessModulus of convexity p-Convexity of operators Banach space Let X and Y be Banach spaces and T : Y → X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T -modulus δ T of convexity and the T -modulus ρ T of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel-Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.

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“…where we use the triangle inequality, the definition of S ′ and (19). A similar calculation holds if z ∈ S ′ and y ∈ S ′′ .…”

Section: 2mentioning

confidence: 98%

Re-examination of Bregman functions and new properties of their divergences

2018

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The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have "Bregmanian properties", can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.

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Uniformly convex functions on Banach spaces

Cited by 39 publications

References 13 publications

Greedy Approximation in Convex Optimization

Greedy Approximation in Convex Optimization

More on convexity and smoothness of operators

Re-examination of Bregman functions and new properties of their divergences

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