Numerical optimization

Abstract

Numerical optimization is introduced as the mathematical foundation for this book, focusing on two basic unconstrained optimization algorithms: line search and trust-region methods. Line search optimization methods are relatively simple and commonly used gradient descent based methods. Their strength lies in their simplicity and ease of implementation, but their convergence properties degrade as the nonlinearity and complexity of the function to be optimized increases. Trust-region, or “restricted step” methods are presented as an often more practical alternative to line search that involved the construction of an approximation, or “model,” of the function to be minimized, together with a dynamic estimate of the region where this model is sufficiently valid. There are a large number of optimization methods, and the interested reader is referred to the bibliographical citations presented in this chapter. In this book we restrict our attention mainly to line search and trust-region methods, since we will use them in the context of their application to extremum seeking control and stability proofs thereof.