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Introduction to nonsmooth optimization. Theory, practice and software. (English) Zbl 1312.90053
Cham: Springer (ISBN 978-3-319-08113-7/hbk; 978-3-319-08114-4/ebook). xviii, 372 p. (2014).
Nonsmooth optimization (NSO) refers to the general problem of minimizing (or maximizing) functions that are typically not differentiable at their minimizers (maximizers). These kinds of problems can be found in many applied fields and real-world modeling systems, for instance, in image denoising, optimal control, neural network training, data mining, economics, and computational chemistry and physics. The book has three parts: Part I deals with the nonsmooth theory: convex and nonconvex analysis with many numerical examples and illustrative figures; nonsmooth optimality conditions from both analytical and geometrical viewpoints; a generalized concept of convexity for nonsmooth functions; brief surveys of different generalizations of subdifferentials and approximations to subdifferentials. Part II deals with nonsmooth optimization (NSO) problems: some real-life NSO problems, for instance, the molecular distance geometry problem, protein structural alignment, data mining, hemivariational inequalities, the power unit-commitment problem, image restoration, and the nonlinear income tax problem; some formulations which lead to NSO problems even though the original problem is smooth; the maximum eigenvalue peroblem; finally, a comprehensive list of test problems used in NSO is given. Part III is a guide to NSO software: short descriptions and the pseudo-codes of the best known methods for the NSO, for instance, subgradient methods, cutting plane methods, bundle methods, gradient sampling method, some hybrid methods, discrete gradient methods; constrained NSO problems; comparisons of implementations of different NSO problems for solving unconstrained problems; finally, a table enabling the quick selection of suitable software for different types of NSO problems. The book is ideal for anyone teaching or attending courses in NSO. It is also well suited for self-access learning for practitioners already familiar with the basics of optimization. It also can serve as a reference text for anyone – including experts – dealing with NSO.

MSC:
90C25 Convex programming
90C26 Nonconvex programming, global optimization
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C56 Derivative-free methods and methods using generalized derivatives
Software:
PNEW; Matlab; LDGB; LMBM; QSM; PBNCGC; DGM
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