scholarly journals Algebraic hyper-structures associated to convex analysis and applications

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 55-65 ◽  
Author(s):  
Delavar Khalafi ◽  
Bijan Davvaz
Keyword(s):  
Convex Programming ◽  
Convex Functions ◽  
Convex Analysis ◽  
Linear Functions

In this paper, we generalize some concepts of convex analysis such as convex functions and linear functions on hyper-structures. Based on new definitions we obtain some important results in convex programming. A few suitable examples have been given for better understanding.

scholarly journals The Engulfing Property from a Convex Analysis Viewpoint

2020 ◽  
Vol 187 (2) ◽  
pp. 408-420 ◽  
Author(s):  
Andrea Calogero ◽  
Rita Pini
Keyword(s):  
Simple Proof ◽  
Convex Functions ◽  
Convex Analysis ◽  
Strict Convexity

Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

scholarly journals Convergence Analysis of Alternating Direction Method of Multipliers for a Class of Separable Convex Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zehui Jia ◽  
Ke Guo ◽  
Xingju Cai
Keyword(s):  
Convergence Analysis ◽  
Convex Programming ◽  
Alternating Direction Method ◽  
Linear Functions ◽  
Method Of Multipliers ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
The Individual

The purpose of this paper is extending the convergence analysis of Han and Yuan (2012) for alternating direction method of multipliers (ADMM) from the strongly convex to a more general case. Under the assumption that the individual functions are composites of strongly convex functions and linear functions, we prove that the classical ADMM for separable convex programming with two blocks can be extended to the case with more than three blocks. The problems, although still very special, arise naturally from some important applications, for example, route-based traffic assignment problems.

scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

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