Optimization for Learning and Control

4Optimization Theory

Mathematical optimization is an indispensable tool in learning and control. Its history goes back to the early seventeenth century with the work by Pierre de Fermat who obtained calculus‐based formulae for identifying optima. This work was later developed further by Joseph‐Louis Lagrange. In this chapter, we will present the foundations of optimization theory. We will start by defining what constitutes an optimization problem and introduce some basic concepts and terminology. We will also introduce the notion of convexity, which allows us to distinguish between convex and general nonlinear optimization problems. The motivation behind this distinction is that convex problems are, roughly speaking, easier to solve than general nonlinear ones. We will pay special attention to properties that are useful for recognizing convexity. Finally, we will also discuss the concept of duality and see how it is used to derive optimality conditions for optimization problems. In Chapter 6, we will see how duality also plays an important role in some optimization methods.

4.1 Basic Concepts and Terminology

Let be a function , where is the domain of and is its codomain. The ...