Upper Semi-Continuity of Subdifferential Mappings

1980 ◽  
Vol 23 (1) ◽  
pp. 11-19 ◽  
Author(s):  
David A. Gregory
Keyword(s):  
Convex Function ◽  
Continuous Convex Function ◽  
Subdifferential Mapping

AbstractCharacterizations of the upper semi-continuity of the subdifferential mapping of a continuous convex function are given.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

scholarly journals Second-order normal vectors to a convex epigraph

1994 ◽  
Vol 50 (1) ◽  
pp. 123-134 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Function ◽  
Mathematical Object ◽  
Second Order ◽  
Normal Vector ◽  
Subdifferential Mapping ◽  
Normal Vectors

The second–order behaviour of a nonsmooth convex function f is reflected by the so–called second–order subdifferential mapping ∂2f. This mathematical object has been intensively studied in recent years. Here we study ∂2f in connection with the geometric concept of “second-order normal vector” to the epigraph of f.

scholarly journals Another Aspect of Triangle Inequality

2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Kichi-Suke Saito ◽  
Runling An ◽  
Hiroyasu Mizuguchi ◽  
Ken-Ichi Mitani
Keyword(s):  
Convex Function ◽  
Triangle Inequality ◽  
Unit Interval ◽  
Usual Sense ◽  
Continuous Convex Function

We introduce the notion of ψ-norm by considering the fact that an absolute normalized norm on C2 corresponds to a continuous convex function ψ on the unit interval [0,1] with some conditions. This is a generalization of the notion of q-norm introduced by Belbachir et al. (2006). Then we show that a ψ-norm is a norm in the usual sense.

scholarly journals Hausdorff dimension of the nondifferentiability set of a convex function

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5827-5831
Author(s):  
Reza Mirzaie
Keyword(s):  
Riemannian Manifold ◽  
Hausdorff Dimension ◽  
Convex Function ◽  
Open Subset ◽  
Upper Bound ◽  
Continuous Convex Function

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ? 2, has Hausdorff dimension at most n - 2.

scholarly journals A continuity property related to an index of non-separability and its applications

1992 ◽  
Vol 46 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Warren B. Moors
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Function ◽  
Compact Subset ◽  
Metric Space ◽  
Dense Subset ◽  
Countable Family ◽  
Set Valued Mapping ◽  
Continuous Convex Function ◽  
Fréchet Differentiable

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

Matrices Doubly Stochastic by Blocks

1977 ◽  
Vol 29 (3) ◽  
pp. 559-577 ◽  
Author(s):  
Pal Fischer ◽  
John A. R. Holbrook
Keyword(s):  
Convex Function ◽  
Classical Result ◽  
Doubly Stochastic ◽  
Continuous Convex Function ◽  
Image Position

The present work stems from the following classical result, due to G. H. Hardy, J. E. Littlewood, G. Pólya [7], and R. Rado [10].THEOREM 1. Concerning a pair of n-tuples x, y ϵ Rn, the following four statementsare equivalent:(a) for every continuous, convex function f : R → R

scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
Author(s):  
P. Kanniappan
Keyword(s):  
Convex Function ◽  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

A Generalization of an Inequality of Bhattacharya and Leonetti

1996 ◽  
Vol 39 (4) ◽  
pp. 438-447 ◽  
Author(s):  
Ritva Hurri-Syrjänen
Keyword(s):  
Convex Function ◽  
Sobolev Class ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Continuous Convex Function ◽  
John Domains ◽  
Image Position

AbstractWe show that bounded John domains and bounded starshaped domains with respect to a point satisfy the following inequalitywhere F: [0, ∞) → [0, ∞) is a continuous, convex function with F(0) = 0, and u is a function from an appropriate Sobolev class. Constants b and K do depend at most on D. If F(x) = xp, 1 ≤ p < ∞, this inequality reduces to the ordinary Poincaré inequality.

scholarly journals A duality theorem for nondifferentiable convex programming with operatorial constraints

1980 ◽  
Vol 22 (1) ◽  
pp. 145-152 ◽  
Author(s):  
P. Kanniappan ◽  
Sundaram M.A. Sastry
Keyword(s):  
Convex Function ◽  
Objective Function ◽  
Programming Problem ◽  
Duality Theorem ◽  
Topological Linear Space ◽  
Lower Semi ◽  
Convex Objective Function ◽  
Continuous Convex Function ◽  
Topological Linear ◽  
Locally Convex

A duality theorem of Wolfe for non-linear differentiable programming is now extended to minimization of a non-differentiable, convex, objective function defined on a general locally convex topological linear space with a non-differentiable operatorial constraint, which is regularly subdifferentiable. The gradients are replaced by subgradients. This extended duality theorem is then applied to a programming problem where the objective function is the sum of a positively homogeneous, lower semi continuous, convex function and a subdifferentiable, convex function. We obtain another duality theorem which generalizes a result of Schechter.

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