This book is intended to be used in 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 investigated. The theories of constrained optimization, convexity and duality are developed rigorously and presented with focus on solving concrete optimization problems. There are plenty of solved examples as well as exercises to which answers and in many cases complete solutions are provided.