UNITEXT175 Theory and Methods of Optimisation

This book originates from the graduate course Theory and Methods of Optimisation taught at the University of Pisa and is primarily intended for students seeking a rigorous yet accessible introduction to optimisation techniques. This book originates from the graduate course Theory and Methods of Optimisation taught at the University of Pisa and is primarily intended for students seeking a rigorous yet accessible introduction to optimisation techniques. While designed with graduate students in mind, the text is largely self-contained and may also be approached by motivated undergraduates with a solid foundation in mathematical analysis, linear algebra, and the basic topology of Euclidean spaces. Key results from differential calculus and topology are recalled throughout, ensuring that the material remains accessible without compromising mathematical depth. Structured in three parts, the text offers a coherent progression from foundational theory to algorithmic methods. The first part provides an introduction to convex analysis; the second covers the theory of linear and nonlinear programming; and the third presents key classical algorithms, including the simplex method and gradient-based techniques. Each chapter builds on previous material, with methods presented in detail, including pseudocode and full convergence proofs. Throughout, the book combines theoretical rigour with applied insight. Every result is proved, and numerous worked examples illustrate the methods in action. This dual emphasis gives the work the character of both a rigorous theoretical text and a practical guide to mathematical optimisation. The book serves both as an introduction and as a comprehensive reference for those interested in applying mathematical models to real-world problems. It will be especially valuable to young researchers in applied mathematics looking to understand the theoretical underpinnings of optimisation methods as well as to those working on the practical implementation of such techniques.