Classical and Modern Optimization(Advanced Textbooks in Mathematics)



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388 pp
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The quest for the optimal is ubiquitous in nature and human behavior. The field of mathematical optimization has a long history and remains active today, particularly in the development of machine learning.Classical and Modern Optimization presents a self-contained overview of classical and modern ideas and methods in approaching optimization problems. The approach is rich and flexible enough to address smooth and non-smooth, convex and non-convex, finite or infinite-dimensional, static or dynamic situations. The first chapters of the book are devoted to the classical toolbox: topology and functional analysis, differential calculus, convex analysis and necessary conditions for differentiable constrained optimization. The remaining chapters are dedicated to more specialized topics and applications.Valuable to a wide audience, including students in mathematics, engineers, data scientists or economists, Classical and Modern Optimization contains more than 200 exercises to assist with self-study or for anyone teaching a third- or fourth-year optimization class.Key Features: oMost textbooks on optimization focus on a special type of problems. This book gives a self-contained overview (covering finite or infinite-dimensional, convex or nonconvex, smooth or non-smooth, static or dynamic problems, iterative methods ...)oThis self-contained book starts with a rigorous but flexible toolbox. It then develops more specialized topics and applications: problems depending on a parameter, dynamic programming in discrete and continuous time, calculus of variations, convex duality (including duality for linear and SD programming), optimal transport, iterative convex minimization methods, applications of optimization to data sciencesoThe book contains more than 200 exercises which may also make it useful to anyone teaching a third/fourth year optimization classoThe book contains some results which are important and useful but not easy to find in textbooks (e.g. generic uniqueness of extremizers, Ekeland’s local surjection theorem, a detailed treatment of first and second-order optimality conditions for constrained minimizers in Banach spaces, the so-called envelope theorem, the subdifferential of a supremum of convex functions, Birkhoff-von Neumann theorem, Nesterov’s accelerated gradient method, convergence of the Douglas-Rachford algorithm, sparse solutions of basis pursuit ...)
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