scholarly journals Generic Fréchet Differentiability of Convex Functions Dominated by a Lower Semicontinuous Convex Function

1998 ◽  
Vol 225 (2) ◽  
pp. 389-400 ◽  
Author(s):  
Cheng Lixin ◽  
Shi Shuzhong ◽  
Wang Bingwu ◽  
E.S Lee
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

Monotone operators and dentability

1978 ◽  
Vol 18 (1) ◽  
pp. 77-82 ◽  
Author(s):  
S.P. Fitzpatrick
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
General Result ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Monotone Operators ◽  
Special Cases ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

P.S. Kenderov has shown that every monotone operator on an Asplund Banach space is continuous on a dense Gδ subset of the interior of its domain. We prove a general result which yields as special cases both Kenderov's Theorem and a theorem of Collier on the Fréchet differentiability of weak* lower semicontinuous convex functions.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

scholarly journals Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui
Keyword(s):  
Convex Function ◽  
Banach Spaces ◽  
Convex Functions ◽  
Convex Set ◽  
Dense Subset ◽  
Lower Semicontinuous ◽  
Asplund Space ◽  
Bounded Subset ◽  
Exposed Point ◽  
Closed Convex Set

In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎. We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.

Subdifferentials Whose Graphs Are Not Norm × Weak* Closed

2003 ◽  
Vol 46 (4) ◽  
pp. 538-545 ◽  
Author(s):  
Jonathan Borwein ◽  
Simon Fitzpatrick ◽  
Roland Girgensohn
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Convex Functions ◽  
Normal Cone ◽  
Separable Hilbert Space ◽  
Lower Semicontinuous ◽  
Weak Topologies

AbstractIn this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

scholarly journals Dual characterizations of relative continuity of convex functions

2001 ◽  
Vol 70 (2) ◽  
pp. 211-224
Author(s):  
J. Benoist ◽  
A. Daniilidis
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Lower Semicontinuous

AbstractVarious properties of continuity for the class of lower semicontinuous convex functions are considered and dual characterizations are established. In particular, it is shown that the restriction of a lower semicontinuous convex function to its domain (respectively, domain of subdifferentiability) is continuous if and only if its subdifferential is strongly cyclically monotone (respectively, σ-cyclically monotone).

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

scholarly journals A note on generalized convex functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Syed Zaheer Ullah ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Generalized Convex Functions

Abstract In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$(η1,η2)-convex function and establish its Hermite–Hadamard type inequality.

scholarly journals A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 890
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Image Processing ◽  
Minimization Problem ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Image Inpainting ◽  
Constrained Minimization ◽  
Convex Minimization Problem ◽  
Mild Conditions ◽  
Constrained Minimization Problem ◽  
Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

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