Estimating convexifiers in continuous optimization

Every function of several variables with the continuous second derivative can be convexified (i.e., made convex) by adding to it a quadratic "convexifier". In this paper we give simple estimates on the bounds of convexifiers. Using the idea of convexification, many problems in applied mathematics can be reduced to convex mathematical programming models. This is illustrated here for nonlinear programs and systems of nonlinear equations

Parametric programming: An illustrative mini encyclopedia

Parametric programming is one of the broadest areas of applied mathematics. Practical problems, that can be described by parametric programming, were recorded in the rock art about thirty millennia ago. As a scientific discipline, parametric programming began emerging only in the 1950\u27s. In this tutorial we introduce, briefly study, and illustrate some of the elementary notions of parametric programming. This is done using a limited theory (mainly for linear and convex models) and by means of examples, figures, and solved real-life case studies. Among the topics discussed are stable and unstable models, such as a projectile motion model (maximizing the range of a projectile), bilevel decision making models and von Stackelberg games of market economy, law of refraction and Snell\u27s law for the ray of light, duality, Zermelo\u27s navigation problems under the water, restructuring in a textile mill, ranking of efficient DMU (university libraries) in DEA, minimal resistance to a gas flow, and semi-abstract parametric programming models. Some numerical methods of input optimization are mentioned and several open problems are posed

Millisecond solar radio bursts in the metric wavelength range

A study and classification of super-short structures (SSSs) recorded during metric type IV bursts is presented. The most important property of SSSs is their duration, at half power ranging from 4-50 ms, what is up to 10 times shorter than spikes at corresponding frequencies. The solar origin of the SSSs is confirmed by one-to-one correspondence between spectral recordings of Artemis-IV1 and high time resolution single frequency measurements of the TSRS2. We have divided the SSSs in the following categories: 1. Broad-Band SSSs: They were partitioned in two subcategories, the SSS-Pulses and Drifting SSSs; 2. Narrow-band: They appear either as Spike-Like SSSs or as Patch-Like SSSs; 3. Complex SSS: They consist of the absorption-emission segments and were morphologically subdivided into Rain-drop Bursts (narrow-band emission head and a broad-band absorption tail) and Blinkers.Comment: Recent Advances in Astronomy and Astrophysics: 7th International Conference of the Hellenic Astronomical Society. AIP Conference Proceedings, Volume 848, pp. 224-228 (2006

Jensen\u27s inequality for nonconvex functions

Jensen\u27s inequality is formulated for convexifiable (generally nonconvex) functions

Modelling with twice continuously differentiable functions

Many real life situations can be described using twice continuously differentiable functions over convex sets with interior points. Such functions have an interesting separation property: At every interior point of the set they separate particular classes of quadratic convex functions from classes of quadratic concave functions. Using this property we introduce new characterizations of the derivative and its zero points. The results are applied to the study of sensitivity of the Cobb-Douglas production function. They are also used to describe the least squares solutions in linear and nonlinear regression

Convexifiable functions in integral calculus

A function is said to be convexifiable if it becomes convex after adding to it a strictly convex quadratic term. In this paper we extend some of the basic integral properties of convex functions to Lipschitz continuously differentiable functions on real line. In particular, we give estimates of the mean value, a "nonstandard" form of Jensen\u27s inequality, and an explicit representation of analytic functions. It is also outlined how one can use convexification to study ordinary differential equations

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 Fermats 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

Characterizing fixed points

A set of sufficient conditions which guarantee the existence of a point x⋆ such that f(x⋆) = x⋆ is called a "fixed point theorem". Many such theorems are named after well-known mathematicians and economists. Fixed point theorems are among most useful ones in applied mathematics, especially in economics and game theory. Particularly important theorem in these areas is Kakutani\u27s fixed point theorem which ensures existence of fixed point for point-to-set mappings, e.g., [2, 3, 4]. John Nash developed and applied Kakutani\u27s ideas to prove the existence of (what became known as) "Nash equilibrium" for finite games with mixed strategies for any number of players. This work earned him a Nobel Prize in Economics that he shared with two mathematicians. Nash\u27s life was dramatized in the movie "Beautiful Mind" in 2001. In this paper, we approach the system f(x) = x differently. Instead of studying existence of its solutions our objective is to determine conditions which are both necessary and sufficient that an arbitrary point x⋆ is a fixed point, i.e., that it satisfies f(x⋆) = x⋆. The existence of solutions for continuous function f of the single variable is easy to establish using the Intermediate Value Theorem of Calculus. However, characterizing fixed points x⋆, i.e., providing answers to the question of finding both necessary and sufficient conditions for an arbitrary given x⋆ to satisfy f(x⋆) = x⋆, is not simple even for functions of the single variable. It is possible that constructive answers do not exist. Our objective is to find them. Our work may require some less familiar tools. One of these might be the "quadratic envelope characterization of zero-derivative point" recalled in the next section. The results are taken from the author\u27s current research project "Studying the Essence of Fixed Points". They are believed to be original. The author has received several feedbacks on the preliminary report and on parts of the project which can be seen on Internet [9]

The fundamental theorem of calculus for Lipschitz functions

Every smooth function in several variables with a Lipschitz derivative, when considered on a compact convex set, is the difference of a convex function and a convex quadratic function. We use this result to decompose anti - derivatives of continuous Lipschitz functions and augment the fundamental theorem of calculus. The augmentation makes it possible to convexify and monotonize ordinary differential equations and obtain possibly new results for integrals of scalar functions and for line integrals. The result is also used in linear algebra where new bounds for the determinant and the spectral radius of symmetric matrices are obtained