Convex Functionals on Convex Sets and Convex Analysis

1985 ◽  
pp. 379-406
Author(s):  
Eberhard Zeidler
Keyword(s):  
Convex Analysis ◽  
Convex Sets ◽  
Convex Functionals

Convex Analysis

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Convex Analysis ◽  
Convex Sets ◽  
Optimization Problems ◽  
Fundamental Concept ◽  
Concave Functions ◽  
Basic Properties ◽  
The Subject ◽  
Theoretical Results

This chapter discusses the elements of convex analysis which are very important in the study of optimization problems. In section 2.1 the fundamentals of convex sets are discussed. In section 2.2 the subject of convex and concave functions is presented, while in section 2.3 generalizations of convex and concave functions are outlined. This section introduces the fundamental concept of convex sets, describes their basic properties, and presents theoretical results on the separation and support of convex sets.

STABILITY OF THE SOLUTION SET OF NON-COERCIVE VARIATIONAL INEQUALITIES

2002 ◽  
Vol 04 (01) ◽  
pp. 145-160 ◽  
Author(s):  
SAMIR ADLY ◽  
MICHEL THÉRA ◽  
EMIL ERNST
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Convex Analysis ◽  
Convex Sets ◽  
Solution Set ◽  
Small Perturbations ◽  
The Stability ◽  
Stability Of The Solution

In this paper, we study the stability of the solution set of a non-coercive variational inequality with respect to small perturbations of the data involved in the problem. This research is done using well-known tools of convex analysis and the concept of well-positioned convex sets (which is defined and studied).

Approximation De Fonctions Convexes Sur Un Espace De Mesures Et Applications

1982 ◽  
Vol 25 (4) ◽  
pp. 392-413 ◽  
Author(s):  
R. Temam
Keyword(s):  
Calculus Of Variations ◽  
Convex Analysis ◽  
Variational Problems ◽  
Existence Theory ◽  
Smooth Functions ◽  
Perfect Plasticity ◽  
Basic Properties ◽  
Convex Functionals

AbstractIn the first part of this article we recall the definition and a few basic properties of convex functionals defined on a space of bounded measures. In the second part we show several results of approximation of the following type: Although a measure μ cannot be approximated in the sense of the norm by smooth functions, we can find an appropriate sequence of smooth functions which converge weakly to the measure μ, the corresponding value of the functional converging to the value of the functional at μ.This article is part of a series on the existence theory of solution of variational problems of mechanics (perfect plasticity), which is based on a systematic utilization of the methods of convex analysis and the calculus of variations.

Differentials and Convex Analysis

2020 ◽  
pp. 7-64
Author(s):  
Robert G. Chambers
Keyword(s):  
Convex Analysis ◽  
Duality Theorem ◽  
Convex Sets ◽  
Directional Derivatives ◽  
Conjugate Duality ◽  
Support Functions ◽  
Essential Properties ◽  
Global Optima ◽  
Maximum Theorem ◽  
The Relationship

Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.

scholarly journals Radon’s and Helly’s theorems for B-1 - convex sets

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 731-735
Author(s):  
Serap Kemali ◽  
Ilknur Yesilce ◽  
Gabil Adilov
Keyword(s):  
Convex Analysis ◽  
Convex Sets ◽  
Important Place

Helly?s, Radon?s, and Caratheodory?s theorems are the basic theorems of convex analysis and have an important place. These theorems have been studied by different authors for different classes of convexity. Caratheodory?s theorem for B-1 - convex sets has been proved before by Adilov and Ye?ilce. In this article, Helly?s and Radon?s theorems are discussed and examined for these sets.

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

Sign in / Sign up

Export Citation Format

Share Document