A method of conjugate subgradients for minimizing nondifferentiable functions

1975 ◽  
pp. 145-173 ◽  
Author(s):  
Philip Wolfe
Keyword(s):  
Nondifferentiable Functions

scholarly journals Nonsmooth invexity

1990 ◽  
Vol 42 (3) ◽  
pp. 437-446 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Lipschitz Functions ◽  
Constrained Optimisation ◽  
Nondifferentiable Functions ◽  
Analytic Tool

The concept of invexity is extended to nondifferentiable functions. Characterisations of nonsmooth invexity are derived as well as results in unconstrained and constrained optimisation and duality. The principal analytic tool is the generalised gradient of Clarke for Lipschitz functions.

Estimating Curvature of Nondifferentiable Functions and Complex Shape Contours

2005 ◽  
Vol 101 (2) ◽  
pp. 362-364 ◽  
Author(s):  
Noah Schwartz
Keyword(s):  
Visual Experience ◽  
Complex Shape ◽  
Simple Modification ◽  
Nondifferentiable Functions ◽  
Point Of Interest ◽  
Complex Shapes ◽  
Psychophysical Analysis ◽  
Sharp Corners

Salvadori and Luccio demonstrated that, for the purpose of psychophysical analysis, the curvature of a point on a continuous single-valued function can be quantified as the inverse ray of the circle osculating the function at the point of interest. While this method is mathematically sound, it does not allow for estimation of curvature in complex shapes containing discontinuities and irregularities such as sharp corners, closed contours, or line intersections—features that commonly occur in normal visual experience. In this paper, a simple modification of this algorithm is presented which overcomes these limitations by using discrete rather than continuous contour sampling, thereby allowing estimation of curvature for a wider variety of shape contours.

scholarly journals Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Li Chen ◽  
Yang Zhao ◽  
Hossein Jafari ◽  
J. A. Tenreiro Machado ◽  
Xiao-Jun Yang
Keyword(s):  
Iteration Method ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Nondifferentiable Functions ◽  
Independent Variables ◽  
Poisson Equations ◽  
Variational Iteration ◽  
Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

scholarly journals Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shu Xu ◽  
Xiang Ling ◽  
Carlo Cattani ◽  
Gong-Nan Xie ◽  
Xiao-Jun Yang ◽  
...  
Keyword(s):  
Heat Flow ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Previous Method ◽  
Heat Equations ◽  
Nondifferentiable Functions ◽  
Variational Iteration ◽  
Nonhomogeneous Heat

The local fractional Laplace variational iteration method is used for solving the nonhomogeneous heat equations arising in the fractal heat flow. The approximate solutions are nondifferentiable functions and their plots are also given to show the accuracy and efficiency to implement the previous method.

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

scholarly journals Invex functions and duality

1985 ◽  
Vol 39 (1) ◽  
pp. 1-20 ◽  
Author(s):  
B. D. Craven ◽  
B. M. Glover
Keyword(s):  
Optimality Condition ◽  
Lagrange Multipliers ◽  
Nondifferentiable Functions ◽  
Duality Result ◽  
Invex Functions ◽  
Abstract Spaces ◽  
Convex Property

AbstractFor both differentiable and nondifferentiable functions defined in abstract spaces we characterize the generalized convex property, here called cone-invexity, in terms of Lagrange multipliers. Several classes of such functions are given. In addition an extended Kuhn-Tucker type optimality condition and a duality result are obtained for quasidifferentiable programming problems.

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