scholarly journals Characterizing approximate global minimizers of the difference of two abstract convex functions with applications

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2431-2445
Author(s):  
A.R. Sattarzadeh ◽  
H. Mohebi
Keyword(s):  
Mild Condition ◽  
Convex Functions ◽  
Global Minimum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximal Elements ◽  
Global Minimizers ◽  
Abstract Subdifferential ◽  
The Difference ◽  
Necessary And Sufficient

In this paper, we first investigate characterizations of maximal elements of abstract convex functions under a mild condition. Also, we give various characterizations for global "-minimum of the difference of two abstract convex functions and, by using the abstract Rockafellar?s antiderivative, we present the abstract ?-subdifferential of abstract convex functions in terms of their abstract subdifferential. Finally, as an application, a necessary and sufficient condition for global ?-minimum of the difference of two increasing and positively homogeneous (IPH) functions is presented.

Equicontinuity of Families of Convex and Concave-Convex Operators

1984 ◽  
Vol 36 (5) ◽  
pp. 883-898 ◽  
Author(s):  
Mohamed Jouak ◽  
Lionel Thibault
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Functions ◽  
Positive Cone ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Convex Operator ◽  
Necessary And Sufficient ◽  
Convex Operators

J. M. Borwein has given in [1] a practical necessary and sufficient condition for a convex operator to be continuous at some point. Indeed J. M. Borwein has proved in his paper that a convex operator with values in an order topological vector space F (with normal positive cone F+) is continuous at some point if and only if it is bounded from above by a mapping which is continuous at this point. This result extends a previous one by M. Valadier in [16] asserting that a convex operator is continuous at a point whenever it is bounded from above by an element in F on a neighbourhood of the concerned point. Note that Valadier's result is necessary if and only if the topological interior of F+ is nonempty. Obviously both results above are generalizations of the classical one about real-valued convex functions formulated in this context exactly as Valadier's result (see for example [5]).

An Oscillation Result for Singular Neutral Equations

1994 ◽  
Vol 37 (1) ◽  
pp. 54-65
Author(s):  
István Gyori ◽  
Janos Turi
Keyword(s):  
Large Class ◽  
Difference Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Equations ◽  
Oscillation Result ◽  
The Difference ◽  
Necessary And Sufficient

AbstractIn this paper, extending the results in [ 1 ], we establish a necessary and sufficient condition for oscillation in a large class of singular (i.e., the difference operator is nonatomic) neutral equations.

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

Lower estimates related to the twisted Dedekind function

2016 ◽  
Vol 12 (07) ◽  
pp. 1801-1811
Author(s):  
Jerzy Kaczorowski ◽  
Kazimierz Wiertelak
Keyword(s):  
Dirichlet Character ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Difference Formula ◽  
The Difference ◽  
Necessary And Sufficient

Let [Formula: see text] be a Dirichlet character. The main goal of this paper is to study oscillations of the difference [Formula: see text] where [Formula: see text] denotes the twisted Dedekind function. We prove that for infinitely many odd characters [Formula: see text] called “good”, we have [Formula: see text], and [Formula: see text] when [Formula: see text] is real. We give a necessary and sufficient condition for [Formula: see text] to be good, and in particular we prove that all odd primitive characters are good. We show also that there are infinitely many moduli [Formula: see text], including all prime powers [Formula: see text], for which all odd characters [Formula: see text] are good.

scholarly journals Good decomposition in the class of convex functions of higher order

2006 ◽  
Vol 80 (94) ◽  
pp. 157-169
Author(s):  
Slobodanka Jankovic ◽  
Tatjana Ostrogorski
Keyword(s):  
Positive Integer ◽  
Convex Functions ◽  
Higher Order ◽  
Sufficient Condition ◽  
Slowly Varying Function ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Class Of Functions ◽  
Good Decomposition

The problems investigated in this article are connected to the fact that the class of slowly varying functions is not closed with respect to the operation of subtraction. We study the class of functions Fk?1, which are nonnegative and i-convex for 0<_ i < k, where k is a positive integer. We present necessary and sufficient condition that guarantee that, no matter how we decompose an additively slowly varying function L ? Fk?1 into a sum L = F + G, F,G ? Fk?1, then necessarily F and G are additively slowly varying.

scholarly journals Geometric Theorems and Application of the DOF Analysis of the Workpiece Based on the Constraint Normal Line

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Guojun Luo ◽  
Xiaohui Wang ◽  
Xianguo Yan
Keyword(s):  
Vertical Plane ◽  
Rigid Bodies ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Normal Line ◽  
Computer Aided ◽  
The Difference ◽  
Instantaneous Center ◽  
Necessary And Sufficient ◽  
Geometric Relationship

Aiming at the problem of judging the degree of freedom (DOF) of the workpiece in the fixture by experience, it is difficult to adapt to the analysis of the DOF of some singular workpieces. The workpiece and the fixture are used as rigid bodies, and the workpiece is allowed to move in the plane or space under the constraints of the fixture positioning point, and a set of geometric theorems for judging the DOF and overconstraint of the workpiece can be derived according to the difference in the position of the instantaneous center of the workpiece speed. The judgment of the DOF and overconstraint of the workpiece is abstracted into rules with universal meaning, which effectively overcomes the limitations of existing methods. The research results show that (1) the DOF and overconstraint of the workpiece in the fixture depend entirely on the number of positioning normal lines of the workpiece and their geometric relationship; (2) the necessary and sufficient condition for limiting the DOF of rotation of the workpiece around a certain axis is that the workpiece has a pair of parallel normal lines in the vertical plane of the axis. Using geometric theorems to judge the DOF of the workpiece is more rigorous, simple, and intuitive, which is convenient for computer-aided judgment and the reasonable layout of the positioning points of the workpiece, which can effectively avoid the misjudgment of the DOF and unnecessary overpositioning when the complex workpiece is combined and positioned on different surfaces. Several examples are used to verify the accuracy of the method and correct unreasonable positioning schemes.

scholarly journals A formula on the approximate subdifferential of the difference of convex functions

1992 ◽  
Vol 45 (1) ◽  
pp. 37-41 ◽  
Author(s):  
J.E. Martínez-legaz ◽  
A. Seeger
Keyword(s):  
Convex Functions ◽  
Global Optimality ◽  
Necessary And Sufficient Condition ◽  
Formula One ◽  
Difference Of Convex Functions ◽  
Approximate Subdifferential ◽  
Difference Of Convex ◽  
Nonconvex Optimisation ◽  
The Difference ◽  
Necessary And Sufficient

We give a formula on the ε−subdifferential of the difference of two convex functions. As a by-product of this formula, one recovers a recent result of Hiriart-Urruty, namely, a necessary and sufficient condition for global optimality in nonconvex optimisation.

scholarly journals The Identification of Convex Function on Riemannian Manifold

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Li Zou ◽  
Xin Wen ◽  
Hamid Reza Karimi ◽  
Yan Shi
Keyword(s):  
Riemannian Manifold ◽  
Convex Function ◽  
Global Minimum ◽  
Directional Derivative ◽  
Inequality Constraints ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Gradient ◽  
Necessary And Sufficient ◽  
Global Minimum Point

The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.

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