Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics, making up the title of the book, reflect the two origins of authors, who belong respectively to the academic world and to that of applications. The approach of their book is very comprehensive without being encyclopaedic: the emphasis is on introducing readers in a gradual and digestible manner to the concepts of convex analysis, their interlinking and their implications, with algorithmic ideas worked in.
Theory is interspersed with application and vice versa; illustrative numerical results are given, and over 170 pictures illustrate and support geometric intuition. Throughout the book, ample comments help the reader further to master the concepts and methods, and to understand the motivations, the difficulties, and the relative significance of results.

In the second printing of these two volumes 305 and 306 the authors have made various local improvements and corrections, and updated the bibliography. The index has been considerably augmented and refined.

Autorentext

Jean-Baptiste Hiriart-Urruty has substantial experience in teaching mathematics at various levels of higher education. He has written several textbooks and exercise books in English and French, including the first volume of Mathematical Tapas (Springer, 2016). The present volume is the fourth published by Springer.

Inhalt
Table of Contents Part I.- I. Convex Functions of One Real Variable.- II. Introduction to Optimization Algorithms.- III. Convex Sets.- IV. Convex Functions of Several Variables.- V. Sublinearity and Support Functions.- VI. Subdifferentials of Finite Convex Functions.- VII. Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory.- VIII. Descent Theory for Convex Minimization: The Case of Complete Information.- Appendix: Notations.- 1 Some Facts About Optimization.- 2 The Set of Extended Real Numbers.- 3 Linear and Bilinear Algebra.- 4 Differentiation in a Euclidean Space.- 5 Set-Valued Analysis.- 6 A Bird's Eye View of Measure Theory and Integration.- Bibliographical Comments.- References.