Basics of Nonlinear Optimization Around the Weierstrass Theorem

This textbook gives an introduction to optimization tools which arise around the Weierstrass theorem about the minimum of a lower semicontinuous function. Starting from a Euclidean space, it moves further into the infinite dimensional setting towards the direct variational method, going through differentiation and introducing relevant background information on the way.

Exercises accompany the text and include observations, remarks, and examples that help understand the presented material. Although some basic knowledge of functional analysis is assumed, covering Hilbert and Banach spaces and the Lebesgue integration, the required background material is covered throughout the text, and literature suggestions are provided. For less experienced readers, a summary of some optimization techniques is also included.

The book will appeal to both students and instructors in specialized courses on optimization, wishing to learn more about variational methods.

- 1. The Weierstrass Theorem - the origin of optimization.- 2. Some basics from functional analysis and function spaces.- 3. Differentiation in infinite dimensional spaces.- 4. On the Weierstrass Theorem in infinite dimensional spaces.- 5. Applications to multiple integrals.

Marek Galewski has been a professor of mathematics at the Institute of Mathematics, Faculty of Technical Physics, Information Technology and Applied Mathematics, Lodz University of Technology since 2010. Between 1998-2010 he worked at the University of Lodz, first as instructor, then as assistant professor and from 2009 as associate professor. He works in nonlinear analysis with emphasis on boundary value problems investigated by variational and monotonicity methods. His current research concentrates on the interplay between variational and monotonicity methods.