This book is intended for a first course treating nonlinear optimization problems. It starts with a review of the necessary concepts of calculus and linear algebra, including positive definite matrices. The classical numerical optimization methods with line-search algorithms are motivated and derived, and some of their properties are investigated. The theories of constrained optimization, convexity and duality are developed rigorously and presented with focus on solving concrete optimization problems. There are numerous solved examples and exercises for which answers and in many cases complete solutions are provided. The second edition includes revisions, improved explanations, and additional content, figures and exercises. The companion book Linear and Combinatorial Optimization — A Basic Course can be read independently.