Optimization is the art, science and mathematics of finding the "best" member of a finite or infinite set of possible choices, based on some objective measure of the merit of each choice in the set. Three key facets of the subject are:

- the construction of optimization models that capture the range of available choices within a feasible set and the measure-of-merit of any particular choice in a feasible set relative to its competitors;

- the invention and implementation of efficient algorithms for solving optimization models;

- a mathematical principle of duality that relates optimization models to one another in a fundamental way. Duality cuts across the entire field of optimization and is useful, in particular, for identifying optimality conditions, i.e., criteria that a given member of a feasible set must satisfy in order to be an optimal solution.

This booklet provides a gentle introduction to the above topics and will be of interest to college students taking an introductory course in optimization, high school students beginning their studies in mathematics and science, the general reader looking for an overall sense of the field of optimization, and specialists in optimization interested in developing new ways of teaching the subject to their students.

John Lawrence Nazareth is Professor Emeritus in the Department of Mathematics at Washington State University and Affiliate Professor in the Department of Applied Mathematics at the University of Washington. He is the author of two recent books also published by Springer-Verlag which explore the above topics in more depth, Differentiable Optimization and Equation Solving (2003) and DLP andExtensions: An Optimization Model and Decision Support System (2001).

Numerical Algorithmic Science and Engineering (NAS&E), or more compactly, Numerical Algorithmics, is the theoretical and empirical study and the practical implementation and application of algorithms for solving finite-dimensional problems of a numeric nature. The variables of such problems are either discrete-valued, or continuous over the reals, or, and as is often the case, a combination of the two, and they may or may not have an underlying network/graph structure. This re-emerging discipline of numerical algorithmics within computer science is the counterpart of the now well-established discipline of numerical analysis within mathematics, where the latter's emphasis is on infinite-dimensional, continuous numerical problems and their finite-dimensional, continuous approximates. A discussion of the underlying rationale for numerical algorithmics, its foundational models of computation, its organizational details, and its role, in conjunction with numerical analysis, in support of the modern modus operandi of scientific computing, or computational science & engineering, is the primary focus of this short monograph. It comprises six chapters, each with its own bibliography. Chapters 2, 3 and 6 present the book's primary content. Chapters 1, 4, and 5 are briefer, and they provide contextual material for the three primary chapters and smooth the transition between them. Mathematical formalism has been kept to a minimum, and, whenever possible, visual and verbal forms of presentation are employed and the discussion enlivened through the use of motivating quotations and illustrative examples. The reader is expected to have a working knowledge of the basics of computer science, an exposure to basic linear algebra and calculus (and perhaps some real analysis), and an understanding of elementary mathematical concepts such as convexity of sets and functions, networks and graphs, and so on. Although this book is not suitable for use as the principal textbook for a course on numerical algorithmics (NAS&E), it will be of value as a supplementary reference for a variety of courses. It can also serve as the primary text for a research seminar. And it can be recommended for self-study of the foundations and organization of NAS&E to graduate and advanced undergraduate students with sufficient mathematical maturity and a background in computing. When departments of computer science were first created within universities worldwide during the middle of the twentieth century, numerical analysis was an important part of the curriculum. Its role within the discipline of computer science has greatly diminished over time, if not vanished altogether, and specialists in that area are now to be found mainly within other fields, in particular, mathematics and the physical sciences. A central concern of this monograph is the regrettable, downward trajectory of numerical analysis within computer science and how it can be arrested and suitably reconstituted. Resorting to a biblical metaphor, numerical algorithmics (NAS&E) as envisioned herein is neither old wine in new bottles, nor new wine in old bottles, but rather this re-emerging discipline is a decantation of an age-old vintage that can hopefully find its proper place within the larger arena of computer science, and at what appears now to be an opportune time.