Riemann’s example of a continuous, ’nondifferentiable’ function

1978 ◽  
Vol 1 (1) ◽  
pp. 40-44 ◽  
Author(s):  
Erwin Neuenschwander
Keyword(s):  
Nondifferentiable Function

On Measures Determined by Continuous Functions that are not of Bounded Variation

1970 ◽  
Vol 13 (1) ◽  
pp. 121-124 ◽  
Author(s):  
J. H. W. Burry ◽  
H. W. Ellis
Keyword(s):  
Continuous Function ◽  
Total Variation ◽  
Bounded Variation ◽  
Real Line ◽  
Open Interval ◽  
Continuous Functions ◽  
Outer Measure ◽  
Function Of Bounded Variation ◽  
Borel Sets ◽  
Nondifferentiable Function

In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.

scholarly journals Copositivity meets D. C. optimization

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Abdellatif Moudafi ◽  
Paul-Emile Mainge
Keyword(s):  
Gradient Method ◽  
Convex Functions ◽  
Symmetric Matrix ◽  
Stationary Points ◽  
Proximal Gradient Method ◽  
Nondifferentiable Function ◽  
Difference Of Convex Functions ◽  
Difference Of Convex

<p style='text-indent:20px;'>Based on a work by M. Dur and J.-B. Hiriart-Urruty[<xref ref-type="bibr" rid="b3">3</xref>], we consider the problem of whether a symmetric matrix is copositive formulated as a difference of convex functions problem. The convex nondifferentiable function in this d.c. decomposition being proximable, we then apply a proximal-gradient method to approximate the related stationary points. Whereas, in [<xref ref-type="bibr" rid="b3">3</xref>], the DCA algorithm was used.</p>

scholarly journals On an algorithm in nondifferential convex optimization

2013 ◽  
Vol 23 (1) ◽  
pp. 59-71
Author(s):  
Nada Djuranovic-Milicic ◽  
Milanka Gardasevic-Filipovic
Keyword(s):  
Convex Optimization ◽  
Objective Function ◽  
Directional Derivative ◽  
Second Order ◽  
Nondifferentiable Function ◽  
Convergence Proof ◽  
Yosida Regularization

In this paper an algorithm for minimization of a nondifferentiable function is presented. The algorithm uses the Moreau-Yosida regularization of the objective function and its second order Dini upper directional derivative. The purpose of the paper is to establish general hypotheses for this algorithm, under which convergence occurs to optimal points. A convergence proof is given, as well as an estimate of the rate of the convergence.

scholarly journals A trust region method using subgradient for minimizing a nondifferentiable function

2009 ◽  
Vol 19 (2) ◽  
pp. 249-262 ◽  
Author(s):  
Milanka Gardasevic-Filipovic
Keyword(s):  
Objective Function ◽  
Necessary Conditions ◽  
Trust Region Method ◽  
Trust Region ◽  
Accumulation Point ◽  
Second Order ◽  
Nondifferentiable Function ◽  
Region Method ◽  
Second Order Necessary Conditions

The minimization of a particular nondifferentiable function is considered. The first and second order necessary conditions are given. A trust region method for minimization of this form of the objective function is presented. The algorithm uses the subgradient instead of the gradient. It is proved that the sequence of points generated by the algorithm has an accumulation point which satisfies the first and second order necessary conditions.

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