This book provides a clear, modern, and application‑driven introduction to mathematical optimization—one of the central technologies underlying today’s data‑driven decision making. Designed for students and practitioners encountering optimization for the first time, the book develops core concepts through intuitive explanations, concrete examples, and algorithmic insights, while maintaining mathematical rigor where it matters.

Beginning with smooth unconstrained problems, the text builds a unified framework connecting calculus, convexity, and optimality conditions. It progresses naturally to constrained and nonsmooth optimization, introducing subgradients, normal cones, and the Karush–Kuhn–Tucker condition in an accessible and coherent manner. Along the way, readers learn how fundamental algorithms—such as gradient, Newton’s, and simplex methods—emerge from the underlying theory and how they behave in practice.

The book places strong emphasis on modeling, guiding readers from real‑world problems to optimization formulations and solution strategies. Topics include linear and quadratic programming, network and integer structures, and optimization under uncertainty. Applications are drawn from engineering, operations research, data science, machine learning, and analytics, illustrating how optimization serves as a unifying language across disciplines.

Written as lecture notes for a one‑semester introductory course, this book assumes only basic knowledge of calculus and linear algebra. It builds intuition, vocabulary, and mathematical comprehension, while largely avoiding formal proofs. Numerous worked examples, exercises, and fully solved problems support self‑study and classroom use.

This book is ideal for advanced undergraduate students, master’s students, and practitioners seeking a principled yet approachable introduction to optimization, as well as a foundation for further study in applied mathematics, operations research, and analytics.

Dr. Johannes O. Royset is professor in the Daniel J. Epstein Department, University of Southern California. He was awarded a National Research Council postdoctoral fellowship in 2003, a Young Investigator Award from the Air Force Office of Scientific Research in 2007, and the Barchi Prize as well as the MOR Journal Award from the Military Operations Research Society in 2009. He received the Goodeve Medal from the Operational Research Society in 2019. Professor Royset was a plenary speaker at the International Conference on Stochastic Programming in 2016, the SIAM Conference on Uncertainty Quantification in 2018, and the INFORMS Security Conference in 2022. He has a PhD from UC Berkeley (2002). Professor Royset has been an associate or guest editor of several journals including SIAM Journal on Optimization, Operations Research, and Mathematical Programming. He has published two books and more than 100 articles.