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Optimization and approximation
Pedregal P., Springer International Publishing, New York, NY, 2017. 254 pp. Type: Book (978-3-319648-42-2)
Date Reviewed: Feb 28 2018

Optimization problems are ubiquitous, and the term alone indicates that it is important. Mathematically, optimization refers to finding the largest or the smallest value that a function can take. For our everyday lives, this can range from simple scenarios like configuring a car under a budget constraint, to more complex ones such as optimally distributing a certain amount of money in the stock market, maximizing traffic flow in a city, or finding a weight configuration in an artificial neural network that minimizes the output error. Surprisingly, most of the situations we encounter in everyday private and corporate life can be translated into optimization problems. People working in the field translate a real-world situation into variables, equations, and formulas and, if necessary, define constraints. A first analysis should already allow one to conclude if a solution to the problem exists and if the cost functional, which is the output one seeks to minimize or maximize, drifts off to infinity. As the author states, if the former is true and the latter is not, the candidates for the extreme values must be detected, and one must verify if the structural conditions on the involved functions ensure that optimality conditions are sufficient for finding the desired solution. These two steps are elaborated in detail in this book.

The book has three sections, comprising mathematical programming, variational problems, and optimal control. It is practically oriented, as each of the sections is introduced with practical and vivid examples, such as a company willing to invest a certain amount of money in machinery, or an optimal growth problem for an aggregate economy. In the first part, readers learn about linear and nonlinear programming, as well as numerical approximation. Even only one occurrence of a simple nonlinear function, such as a square, among millions of linear functions, disqualifies a problem from being linear. It is of utmost importance to understand how problems are correctly formulated and assigned to respective classes, and the author helps readers develop the required intuition using examples and teaches them how to formulate a mathematically accurate problem by providing detailed explanations of the mathematical formalisms. Sometimes, for example, when a problem is time-critical, one may not have the required computational power to one’s avail to find a solution, and sometimes it may not even be possible for a powerful supercomputer to find an optimal solution in finite time. Here, approximation comes into play, where one approximates the optimal solution to find a solution that is “good enough” for a particular situation. Again, the author explains very well how and when to make use of numerical approximation and describes the most commonly used algorithms in detail. The second part of the book is all about variational problems, where extremals for functionals rather than functions must be found. Instead of finite-dimensional vectors, the solution space now consists of infinite-dimensional functions. The third part introduces the reader to optimal control, where functions represent possibilities and strategies, and integrals express costs. One might immediately think about the control of robots in manufacturing, but again, the author uses very vivid examples one may not expect.

Although the book does not go in depth in each of the chapters, it is for people with intermediate mathematical knowledge with an interest in learning about optimization and approximation. The book teaches readers about mathematical programming problems, approximation, and optimal control, with a focus on practical applications. It may be used as the foundation for a course on optimization.

Reviewer:  Florian Neukart Review #: CR145891 (1805-0204)
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