ON THE DERIVATIVE OF SMOOTH MEANINGFUL FUNCTIONS

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.

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