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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 the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books.
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.