Mathematics of Optimization: Smooth and Nonsmooth Case
Mathematics of Optimization: Smooth and Nonsmooth Case

Author: Giorgio Giorgi

Publisher: Elsevier

Published: 2004-03-10

Total Pages: 615

ISBN-13: 008053595X

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The book is intended for people (graduates, researchers, but also undergraduates with a good mathematical background) involved in the study of (static) optimization problems (in finite-dimensional spaces). It contains a lot of material, from basic tools of convex analysis to optimality conditions for smooth optimization problems, for non smooth optimization problems and for vector optimization problems.The development of the subjects are self-contained and the bibliographical references are usually treated in different books (only a few books on optimization theory deal also with vector problems), so the book can be a starting point for further readings in a more specialized literature.Assuming only a good (even if not advanced) knowledge of mathematical analysis and linear algebra, this book presents various aspects of the mathematical theory in optimization problems. The treatment is performed in finite-dimensional spaces and with no regard to algorithmic questions. After two chapters concerning, respectively, introductory subjects and basic tools and concepts of convex analysis, the book treats extensively mathematical programming problems in the smmoth case, in the nonsmooth case and finally vector optimization problems.· Self-contained· Clear style and results are either proved or stated precisely with adequate references· The authors have several years experience in this field· Several subjects (some of them non usual in books of this kind) in one single book, including nonsmooth optimization and vector optimization problems· Useful long references list at the end of each chapter

Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control
Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control

Author: Marko M Makela

Publisher: World Scientific

Published: 1992-05-07

Total Pages: 268

ISBN-13: 9814522414

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This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization. The second part is devoted to the methods of nonsmooth optimization and their development. A proximal bundle method for nonsmooth nonconvex optimization subject to nonsmooth constraints is constructed. In the last part nonsmooth optimization is applied to problems arising from optimal control of systems covered by partial differential equations. Several practical problems, like process control and optimal shape design problems are considered.

Nonsmooth Analysis
Nonsmooth Analysis

Author: Winfried Schirotzek

Publisher: Springer Science & Business Media

Published: 2007-05-26

Total Pages: 380

ISBN-13: 3540713336

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This book treats various concepts of generalized derivatives and subdifferentials in normed spaces, their geometric counterparts and their application to optimization problems. It starts with the subdifferential of convex analysis, passes to corresponding concepts for locally Lipschitz continuous functions and then presents subdifferentials for general lower semicontinuous functions. All basic tools are presented where they are needed: this concerns separation theorems, variational and extremal principles as well as relevant parts of multifunction theory. Each chapter ends with bibliographic notes and exercises.

Modern Nonconvex Nondifferentiable Optimization
Modern Nonconvex Nondifferentiable Optimization

Author: Ying Cui

Publisher: Society for Industrial and Applied Mathematics (SIAM)

Published: 2022

Total Pages: 0

ISBN-13: 9781611976731

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"This monograph serves present and future needs where nonconvexity and nondifferentiability are inevitably present in the faithful modeling of real-world applications of optimization"--

Invexity and Optimization
Invexity and Optimization

Author: Shashi K. Mishra

Publisher: Springer Science & Business Media

Published: 2008-05-23

Total Pages: 269

ISBN-13: 3540785612

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Invexity and Optimization presents results on invex function and their properties in smooth and nonsmooth cases, pseudolinearity and eta-pseudolinearity. Results on optimality and duality for a nonlinear scalar programming problem are presented, second and higher order duality results are given for a nonlinear scalar programming problem, and saddle point results are also presented. Invexity in multiobjective programming problems and Kuhn-Tucker optimality conditions are given for a multiobjecive programming problem, Wolfe and Mond-Weir type dual models are given for a multiobjective programming problem and usual duality results are presented in presence of invex functions. Continuous-time multiobjective problems are also discussed. Quadratic and fractional programming problems are given for invex functions. Symmetric duality results are also given for scalar and vector cases.

Optimality Conditions in Convex Optimization
Optimality Conditions in Convex Optimization

Author: Anulekha Dhara

Publisher: CRC Press

Published: 2011-10-17

Total Pages: 446

ISBN-13: 1439868220

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Optimality Conditions in Convex Optimization explores an important and central issue in the field of convex optimization: optimality conditions. It brings together the most important and recent results in this area that have been scattered in the literature—notably in the area of convex analysis—essential in developing many of the important results in this book, and not usually found in conventional texts. Unlike other books on convex optimization, which usually discuss algorithms along with some basic theory, the sole focus of this book is on fundamental and advanced convex optimization theory. Although many results presented in the book can also be proved in infinite dimensions, the authors focus on finite dimensions to allow for much deeper results and a better understanding of the structures involved in a convex optimization problem. They address semi-infinite optimization problems; approximate solution concepts of convex optimization problems; and some classes of non-convex problems which can be studied using the tools of convex analysis. They include examples wherever needed, provide details of major results, and discuss proofs of the main results.

Nonsmooth Optimization
Nonsmooth Optimization

Author: Claude Lemarechal

Publisher: Elsevier

Published: 2014-05-19

Total Pages: 195

ISBN-13: 1483188760

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Nonsmooth Optimization contains the proceedings of a workshop on non-smooth optimization (NSO) held from March 28 to April 8,1977 in Austria under the auspices of the International Institute for Applied Systems Analysis. The papers explore the techniques and theory of NSO and cover topics ranging from systems of inequalities to smooth approximation of non-smooth functions, as well as quadratic programming and line searches. Comprised of nine chapters, this volume begins with a survey of Soviet research on subgradient optimization carried out since 1962, followed by a discussion on rates of convergence in subgradient optimization. The reader is then introduced to the method of subgradient optimization in an abstract setting and the minimal hypotheses required to ensure convergence; NSO and nonlinear programming; and bundle methods in NSO. A feasible descent algorithm for linearly constrained least squares problems is described. The book also considers sufficient minimization of piecewise-linear univariate functions before concluding with a description of the method of parametric decomposition in mathematical programming. This monograph will be of interest to mathematicians and mathematics students.

Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization
Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

Author: Qamrul Hasan Ansari

Publisher: CRC Press

Published: 2013-07-18

Total Pages: 294

ISBN-13: 1439868212

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Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction.The first part of the book focuses on generalized convexity and generalized

Convex Optimization
Convex Optimization

Author: Stephen P. Boyd

Publisher: Cambridge University Press

Published: 2004-03-08

Total Pages: 744

ISBN-13: 9780521833783

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Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.

An Introduction to Nonlinear Optimization Theory
An Introduction to Nonlinear Optimization Theory

Author: Marius Durea

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2014-01-01

Total Pages: 398

ISBN-13: 3110427354

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The goal of this book is to present the main ideas and techniques in the field of continuous smooth and nonsmooth optimization. Starting with the case of differentiable data and the classical results on constrained optimization problems, and continuing with the topic of nonsmooth objects involved in optimization theory, the book concentrates on both theoretical and practical aspects of this field. This book prepares those who are engaged in research by giving repeated insights into ideas that are subsequently dealt with and illustrated in detail.