Lower semicontinuous convex functions

1989 ◽  
pp. 40-63
Author(s):  
Robert R. Phelps
Keyword(s):  
Convex Functions ◽  
Lower Semicontinuous

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.

scholarly journals Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 21
Author(s):  
Yasunori Kimura ◽  
Keisuke Shindo
Keyword(s):  
Asymptotic Behavior ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Convergent Sequence ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Geodesic Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function ◽  
Sequence Of Functions

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey

2003 ◽  
Vol 133 (6) ◽  
pp. 1361-1377 ◽  
Author(s):  
Agnieszka Kałamajska
Keyword(s):  
Differential Operator ◽  
Convex Functions ◽  
Conservation Law ◽  
Necessary Conditions ◽  
Lower Semicontinuous ◽  
Convexity Condition ◽  
Geometric Conditions ◽  
Rank One ◽  
Constant Coefficients ◽  
Lower Semicontinuous Functional

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form If(u) = ∫Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.

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⁎.

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.

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