Numerical Infinities and Infinitesimals in Optimization

This book provides a friendly introduction to the paradigm and proposes a broad panorama of killing applications of the Infinity Computer in optimization: radically new numerical algorithms, great theoretical insights, efficient software implementations, and interesting practical case studies. This is the first book presenting to the readers interested in optimization the advantages of a recently introduced supercomputing paradigm that allows to numerically work with different infinities and infinitesimals on the Infinity Computer patented in several countries. One of the editors of the book is the creator of the Infinity Computer, and another editor was the first who has started to use it in optimization. Their results were awarded by numerous scientific prizes. This engaging book opens new horizons for researchers, engineers, professors, and students with interests in supercomputing paradigms, optimization, decision making, game theory, and foundations of mathematics and computer science. ¿Mathematicians have never been comfortable handling infinities¿ But an entirely new type of mathematics looks set to by-pass the problem¿ Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem¿ ¿ MIT Technology Review ¿These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.¿ A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing. ¿I am sure that the new approach ¿ will have a very deep impact both on Mathematics and Computer Science.¿ D. Trigiante, Computational Management Science. ¿Within the grossone framework, it becomes feasible to deal computationally with infinite quantities, in a way that is both new (in the sense that previously intractable problems become amenable to computation) and natural¿. R. Gangle, G. Caterina, F. Tohmé, Soft Computing.¿The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort.¿ P. Amodio, L. Brugnano, F. Iavernaro & F. Mazzia, Soft Computing