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Croatian Operational Research Review, Vol. 2 No. 1, 2011.

Sažetak sa skupa

ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS

Sanjo Zlobec ; Department of Mathematics and Statistics, McGill University, Montreal, Canada

Puni tekst: engleski, pdf (221 KB) str. 3-3 preuzimanja: 175* citiraj
APA 6th Edition
Zlobec, S. (2011). ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS. Croatian Operational Research Review, 2 (1), 3-3. Preuzeto s https://hrcak.srce.hr/96599
MLA 8th Edition
Zlobec, Sanjo. "ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS." Croatian Operational Research Review, vol. 2, br. 1, 2011, str. 3-3. https://hrcak.srce.hr/96599. Citirano 22.05.2019.
Chicago 17th Edition
Zlobec, Sanjo. "ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS." Croatian Operational Research Review 2, br. 1 (2011): 3-3. https://hrcak.srce.hr/96599
Harvard
Zlobec, S. (2011). 'ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS', Croatian Operational Research Review, 2(1), str. 3-3. Preuzeto s: https://hrcak.srce.hr/96599 (Datum pristupa: 22.05.2019.)
Vancouver
Zlobec S. ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS. Croatian Operational Research Review [Internet]. 2011 [pristupljeno 22.05.2019.];2(1):3-3. Dostupno na: https://hrcak.srce.hr/96599
IEEE
S. Zlobec, "ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS", Croatian Operational Research Review, vol.2, br. 1, str. 3-3, 2011. [Online]. Dostupno na: https://hrcak.srce.hr/96599. [Citirano: 22.05.2019.]

Sažetak
The derivative of a function f in n variables at a point x* is one of the most important tools in mathematical modelling. If this object exists, it is represented by the row n-tuple f(x*) = [∂f/∂xi(x*)] called the gradient of f at x*, abbreviated: “the gradient”. The evaluation of f(x*) is usually done in two stages, first by calculating the n partials and then their values at x = x*. In this talk we give an alternative approach. We show that one can characterize the gradient without differentiation! The idea is to fix an arbitrary row n-tuple G and answer
the following question: What is a necessary and sufficient condition such that G is the gradient of a given f at a given x*? The answer is given after adjusting the quadratic envelope property introduced in [3].
We work with smooth, i.e., continuously differentiable, functions with a Lipschitz derivative on a compact convex set with a non-empty interior. Working with this class of functions is not a serious restriction. In fact,
loosely speaking, “almost all” smooth meaningful functions used in modelling of real life situations are expected to have a bounded “acceleration” hence they belong to this class. In particular, the class contains all twice differentiable functions [1]. An important property of the functions from this class is that every f can be represented as the difference of some convex function and a convex quadratic function. This decomposition was used in [3] to characterize the zero derivative points. There we obtained reformulations and augmentations of some well known classic results on optimality such as Fermats extreme value theorem (known from high school) and the Lagrange multiplier theorem from calculus [2, 3]. In this talk we extend
the results on zero derivative points to characterize the relation G = f(x*), where G is an arbitrary n-tuple. Some special cases: If G = O, we recover the results on zero derivative points. For functions of a single
variable on I = [a, b], the choice G = [f(b) – f(a)]/(b – a) yields characterizations of points c where the instantaneous and average rates of change coincide [4], etc. The celebrated mean value theorem [2] claims that at least one such point c exists but it does not characterize it. These ideas are illustrated by examples and a photograph of an overpass in Beijing. A successful implementation of the new approach requires familiarity with the basic theory of infinite sequences.

[1] Floudas, C. A. and C. E. Gounaris: An overview of advances in global optimization during 2003-2008,” a chapter in the book Lectures on Global Optimization (P. M. Pardalos and T. F. Coleman, editors), Fields Institute Communications, v. 55 (2009) 105-154.
[2] Neralić, L. and B. Šego, B.: Matematika, Element, Zagreb, 2009.
[3] Characterizing zero-derivative points, J. Global Optimization 46 (2010) 155-161. (Published on line: 2 July 2009.)
[4] On the behaviour of functions around zero-derivative points, Int. J. Optimization: Theory, Methods and Applications 1 (2009) 329-340.

Ključne riječi
mathematical modelling; derivative

Hrčak ID: 96599

URI
https://hrcak.srce.hr/96599

Posjeta: 314 *