Optimization in Function Spaces

Amol Sasane

• Publisher: Dover Publications
• Year: 2016
• ISBN:
  9780486810966
• Edition: 1
• Topic: Mathematics
• Subtopic: Optimization
• Index: Mathematics

This highly readable volume on optimization in function spaces is based on author Amol Sasane's lecture notes, which he developed over several years while teaching a course for third-year undergraduates at the London School of Economics. The classroom-tested text is written in an informal but precise style that emphasizes clarity and detail, taking students step by step through each subject.
Numerous examples throughout the text clarify methods, and a substantial number of exercises provide reinforcement. Detailed solutions to all of the exercises make this book ideal for self-study. The topics are relevant to students in engineering and economics as well as mathematics majors. Prerequisites include multivariable calculus and basic linear algebra. The necessary background in differential equations and elementary functional analysis is developed within the text, offering students a self-contained treatment.

Part 1

Calculus in Normed Spaces; Euler-Lagrange Equation

Chapter 1

Calculus in Normed Spaces

As we had discussed in the previous chapter, we wish to differentiate functions living on vector spaces (such as C[a, b]) whose elements are functions, and taking values in

. In order to talk about the derivative, we need a notion of closeness between points of a vector space so that the derivative can be defined. It turns out that vector spaces such as C[a, b] can be equipped with a “norm,” and this provides a “distance” between two vectors. Having done this, we have the familiar setting of calculus, and we can talk about the derivative of a function living on a normed space. We then also have analogues of the two facts from ordinary calculus relevant to optimization that were mentioned in the previous chapter, namely the vanishing of the derivative for minimizers, and the sufficiency of this condition for minimization when the function has an increasing derivative. Thus the outline of this chapter is as follows:

(1) We will learn the notion of a “normed space,” that is a vector space equipped with a “norm,” enabling one to measure distances between vectors in the vector space. This makes it possible to talk about concepts from calculus, and in particular the notion of differentiability of functions between normed spaces.

(2) We will define what we mean by the derivative of a function f : X → Y, where X, Y are normed spaces, at a point x₀ ∈ X. In other words...