scholarly journals Novel View on Classical Convexity Theory

2020 ◽  
Vol 16 (3) ◽  
pp. 291-311
Author(s):  
Vitali Milman ◽  
◽  
Liran Rotem ◽  
Keyword(s):  
Convexity Theory ◽  
Classical Convexity

scholarly journals On the symmetrized s-divergence

2020 ◽  
Vol 18 (1) ◽  
pp. 378-385
Author(s):  
Slavko Simić ◽  
Sara Salem Alzaid ◽  
Hassen Aydi
Keyword(s):  
Basic Properties ◽  
Leibler Divergence ◽  
Convexity Theory ◽  
Relative Divergence

Abstract In this study, we work with the relative divergence of type s,s\in {\mathbb{R}} , which includes the Kullback-Leibler divergence and the Hellinger and χ 2 distances as particular cases. We study the symmetrized divergences in additive and multiplicative forms. Some basic properties such as symmetry, monotonicity and log-convexity are established. An important result from the convexity theory is also proved.

scholarly journals New Integral Inequalities via Generalized Preinvex Functions

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 296
Author(s):  
Muhammad Tariq ◽  
Asif Ali Shaikh ◽  
Soubhagya Kumar Sahoo ◽  
Hijaz Ahmad ◽  
Thanin Sitthiwirattham ◽  
...  
Keyword(s):  
Integral Inequality ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Power Mean ◽  
Science And Engineering ◽  
Special Means ◽  
Algebraic Properties ◽  
Convexity Theory ◽  
Hölder’S Integral Inequality ◽  
Preinvex Functions

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.

scholarly journals On the symmetrized S-divergence

2019 ◽  
Vol 29 ◽  
pp. 01004
Author(s):  
Slavko Simić
Keyword(s):  
Leibler Divergence ◽  
Convexity Theory ◽  
Relative Divergence

In this paper we worked with the relative divergence of type s, s ∈ ℝ, which include Kullback-Leibler divergence and the Hellinger and χ2 distances as particular cases. We give here a study of the sym- metrized divergences in additive and multiplicative forms. Some ba-sic properties as symmetry, monotonicity and log-convexity are estab-lished. An important result from the Convexity Theory is also proved.

A generalization of convex functions via support properties

1976 ◽  
Vol 21 (3) ◽  
pp. 341-361 ◽  
Author(s):  
Aharon Ben-Tal ◽  
Adi Ben-Israel
Keyword(s):  
Convex Functions ◽  
Generalized Convex Functions ◽  
Classical Convexity

AbstractWith respect to a given family of functions F, a function is said to be F-convex, if it is supported, at each point, by some member of F. For particular choices of F one obtains the convex functions and the generalized convex functions in the sense of Beckenbach. F-convex functions are characterized and studied, retaining some essential results of classical convexity.

GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

1993 ◽  
Vol 04 (02) ◽  
pp. 193-202 ◽  
Author(s):  
GUSTAVO CORACH ◽  
HORACIO PORTA ◽  
LÁZARO RECHT
Keyword(s):  
Geometric Mean ◽  
Geodesic Segment ◽  
Finsler Metric ◽  
Natural Structure ◽  
Operator Inequalities ◽  
C Algebra ◽  
Operator Means ◽  
Natural Notion ◽  
Invertible Elements ◽  
Classical Convexity

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.

scholarly journals Convexity preserving mappings in subbase convexity theory

1978 ◽  
Vol 81 (1) ◽  
pp. 76-90
Author(s):  
J. Van Mill ◽  
M. Van De Vel
Keyword(s):  
Convexity Theory

scholarly journals Convexity theory for the term structure equation

2007 ◽  
Vol 12 (1) ◽  
pp. 117-147 ◽  
Author(s):  
Erik Ekström ◽  
Johan Tysk
Keyword(s):  
Term Structure ◽  
Structure Equation ◽  
Convexity Theory

GLOBAL OPTIMIZATION OF LINEAR HYBRID SYSTEMS WITH VARYING TIME EVENTS

2005 ◽  
Vol 15 (02) ◽  
pp. 467-472
Author(s):  
CHA KUN LEE ◽  
PAUL I. BARTON
Keyword(s):  
Hybrid System ◽  
Optimization Problems ◽  
Convex Relaxation ◽  
Continuous Systems ◽  
Global Optimality ◽  
Control Parametrization ◽  
Convexity Theory ◽  
Transition Times ◽  
Deterministic Framework ◽  
Finite Number Of Iterations

Dynamic optimization problems with linear hybrid (discrete/continuous) systems embedded whose transition times vary are inherently nonconvex. For a wide variety of applications, a certificate of global optimality is essential, but this cannot be obtained using conventional numerical methods. We present a deterministic framework for the solution of such problems in the continuous time domain. First, the control parametrization enhancing transform is used to transform the embedded dynamic system from a linear hybrid system with scaled discontinuities and varying transition times into a nonlinear hybrid system with stationary discontinuities and fixed transition times. Next, a recently developed convexity theory is applied to construct a convex relaxation of the original nonconvex problem. This allows the problem to be solved in a branch-and-bound framework that can guarantee the global solution within epsilon optimality in a finite number of iterations.

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