This book presents basic optimization principles and gradient-based algorithms to a general audience, in a brief and easy-to-read form. It enables professionals to apply optimization theory to engineering, physics, chemistry, or business economics.／It is intended that this book be used in senior- to graduate-level semester courses in optimization, as offered in mathematics, engineering, com- puter science and operations research departments. Hopefully this book will also be useful to practising professionals in the workplace. The contents of the book represent the fundamental optimization mate- rial collected and used by the author, over a period of more than twenty years, in teaching Practical Mathematical Optimization to undergradu- ate as well as graduate engineering and science students at the University of Pretoria. The principal motivation for writing this work has not been the teaching of mathematics per se, but to equip students with the nec- essary fundamental optimization theory and algorithms, so as to enable them to solve practical problems in their own particular principal fields of interest, be it physics, chemistry, engineering design or business eco- nomics. The particular approach adopted here follows from the author's own personal experiences in doing research in solid-state physics and in mechanical engineering design, where he was constantly confronted by problems that can most easily and directly be solved via the judicious use of mathematical optimization techniques. This book is, however, not a collection of case studies restricted to the above-mentioned specialized research areas, but is intended to convey the basic optimization princi- ples and algorithms to a general audience in such a way that, hopefully, the application to their own practical areas of interest will be relatively simple and straightforward.