Optimization 1

Abstract

What is Optimization? Optimization involves searching for the "best" element from a given set. The study of the properties of optimal solutions precisely forms the objectives of optimization. It is a branch of applied mathematics and numerical analysis, which has been developing for several years and shows relationships with many other fields of mathematics. This topic examines whether local and global extrema exist for a function of one or more variables, with or without constraints. This document is particularly intended for undergraduate students (L3) in mathemat- ics, in accordance with the curriculum of this program. It serves as a course support rich in exercises and numerical examples on unconstrained optimization. It consists of three chapters. In the first chapter, we recall some concepts of differential calculus and notions of convexity that are useful for the rest of the document. At the end of this chapter, a series of exercises is provided, along with a sample exam question with detailed solutions. The second chapter presents the conditions for existence and uniqueness for a non- linear optimization problem without constraints. We will then present the necessary and sufficient conditions for optimality in the case of a general unconstrained optimization problem and in the convex case. The third chapter is dedicated to algorithms for solving a nonlinear optimization problem without constraints. This chapter concludes with a series of exercises and an exam question without solutions.