This book aims at an innovative approach within the framework of convex analysis and optimization, based on an in-depth study of the behavior and properties of the supremum of families of convex functions. It presents an original and systematic treatment of convex analysis, covering standard results and improved calculus rules in subdifferential analysis. The tools supplied in the text allow a direct approach to the mathematical foundations of convex optimization, in particular to optimality and duality theory. Other applications in the book concern convexification processes in optimization, non-convex integration of the Fenchel subdifferential, variational characterizations of convexity, and the study of Chebychev sets. At the same time, the underlying geometrical meaning of all the involved concepts and operations is highlighted and duly emphasized. A notable feature of the book is its unifying methodology, as well as the novelty of providing an alternative or complementary view to the traditional one in which the discipline is presented to students and researchers. 

This textbook can be used for courses on optimization, convex and variational analysis, addressed to graduate and post-graduate students of mathematics, and also students of economics and engineering. It is also oriented to provide specific background for courses on optimal control, data science, operations research, economics (game theory), etc. The book represents a challenging and motivating development for those experts in functional analysis, convex geometry, and any kind of researchers who may be interested in applications of their work.

Rafael Correa, Mathematical Engineering Degree from the University of Chile (1971), Doctor in Engineering, University of Clermont, France (1974) and Doctor in Mathematics Science, Blaise Pascal University, France (1984). He served as Executive Director of CONICYT, the Chilean National Agency for Scientific Research and Development (1990-1994), as Founder and Director of the Mathematical Modeling Center (CMM) at the University of Chile (2000-2007), and as President of O'Higgins University Chile, since its foundation in 2015. Author of fifty papers on Mathematical Optimization and Variational Analysis.

Abderrahim Hantoute, Doctor in Applied Mathematics from Université Paul Sabatier de Toulouse (2003), Postdoc Fellowship in Alicante and Elche Universities 2004-2007, Associate researcher (at CMM, Universidad de Chile until 2020, and then at Universidad de Alicante). According to MathSciNet: 47 papers, with 316 citations by 182authors.

Marco A. López, Doctor in Mathematics from Valencia University (1973), Full Professor since 1981, currently at Alicante University as Emeritus Professor. Doctor Honoris Causa by the University of Limoges (2012), Honorary Adjunct Professor of Federation University, Australia (2013), as and Corresponding Member of the Real Academia de Ciencias of Spain (2014). Research on mathematical optimization, semi-infinite programming, variational analysis and game theory. According to MathSciNet: 151 papers, with 1788 citations by 590 authors.

1. Introduction.- 2. Preliminaries.- 3. Fenchel-Moreau-Rockafellar theory.- 4. Fundamental topics in convex analysis.- 5. Supremum of convex functions.- 6. The supremum in specific contexts.- 7.  Other subdifferential calculus rules.- 8. Miscellaneous.- 9. Exercises - Solutions.- Index.- Glossary of Notations.- Bibliography.