Cardiff University professor reviews UNN professor's book

The book Numerical Infinities and Infinitesimals in Optimization, edited by Yaroslav Sergeyev, Professor of the UNN Department of Software and Supercomputer Technologies and Renato De Leone, Professor of the University of Camerino (Italy), was published in 2022. A major review written by J. Gillard, Professor at Cardiff University (Cardiff, Wales), appeared in the journal Optimization Letters (Springer).

The volume under review is unusual amongst the panorama of books on optimization. It proposes a series of results dealing with optimization theory and numerical algorithms which use infinities and infinitesimals, entities for centuries considered to be solely the preserve of highly theoretical researchers. Only philosophers, logicians, and specialists in the foundations of mathematics and computer science wrote about such highly abstract subjects. In contrast, the present volume is written by experienced people working for decades in applied mathematics, or more precisely, in optimization. Amongst the authors there are more than 20 internationally known experts in optimization. The editors of the volume are also highly regarded scientists being the former President of the International Society of Global Optimization (Y.D. Sergeyev) and the former President of the Italian Operations Research Society (R. De Leone).

The authors of the volume succeeded in utilising infinite and infinitesimal quantities in their numerical algorithms since the quantities they applied were not traditional abstract symbolic entities but numerical ones, which can be represented on the so-called Infinity Computer. This is the first book presenting to readers interested in optimization the advantages of numerical floating-point infinities and infinitesimals - this is how the review begins.

Professor Gillard presents a detailed description of the contents of the book’s 14 chapters and concludes his review as follows:

In conclusion, this book could have deep impact upon not only local, global, multi-objective optimization and machine learning, but also possibly on applied mathematics more broadly and numerical computation. People interested in new ideas for computer science and its foundations and possibly even the philosophy of mathematics will find this volume interesting, as would those working in theoretical or applied optimization.