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The projected subgradient algorithm in convex optimization. (English) Zbl 1464.90063

SpringerBriefs in Optimization. Cham: Springer (ISBN 978-3-030-60299-4/pbk; 978-3-030-60300-7/ebook). vi, 146 p. (2020).
The behavior of subgradient algorithms for constrained minimization problems in a Hilbert space is studied, searching for better approximating the solution of the problem in the presence of computational errors. The book is rigorously written, and organized taking into account the cursiveness of reading. The long proofs of the theorems are placed in annexes to chapters, in order to emphasize the importance of every result in a generating methodology of studying and solving problems. The book consists of five chapters and a bibliography containing 78 titles of fundamental papers in the field of subgradient optimization methods. Chapter 1, entitled Introduction, contains the presentation of the algorithms that are studied in the book: The Subgradient Projection Algorithm and its version taking into account the computation errors, called Subgradient Projection Algorithm with Computational Errors. An extension of the projected subgradient method for the minimization of convex and nonsmooth functions, under the presence of computational errors is studied in Chapter 2, called Nonsmooth Convex Optimization. The properties of the approximate solutions, their convergence to the solution set and upper bounds are presented. In Chapter 3, entitled Extensions, the author extends the projected subgradient method to nonsmooth convex constrained optimization problems in a Hilbert space. In this framework, the objective function is defined on an open convex set and the set of admissible points is not necessarily convex. The case of an objective function defined on the whole Hilbert space is also discussed. Chapter 4, called Zero-Sum Games with Two Players, the author studies an extension of the projected subgradient method for zero-sum games with two players under the presence of computational errors. Here a quasi-nonexpansive retraction on a feasible set is used instead of the projection on a feasible set. In Chapter 5, called Quasiconvex Optimization, minimization of quasiconvex and nonsmooth functions is approached by means of an extension of the projected subgradient method under presence of computational errors. In all cases, conditions to assess the number of iterations that are necessary to obtain a convenient approximate solution are given.

MSC:

90C25 Convex programming
90C48 Programming in abstract spaces
90C52 Methods of reduced gradient type
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
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