Convex Optimization Theory

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Convex Optimization Theory

Dimitri Bertsekas
Convex Optimization Theory Book PDF

Author : Dimitri Bertsekas
Publisher : Athena Scientific
Page : 256 pages
File Size : 47,9 Mb
Release : 2009-06-01
Category : Mathematics
ISBN : 9781886529311

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Convex Optimization Theory by Dimitri Bertsekas Pdf

An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex functions in terms of points, and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework. This on-line version of the book, includes an extensive set of theoretical problems with detailed high-quality solutions, which significantly extend the range and value of the book. The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory). It is an excellent supplement to several of our books: Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2017), Network Optimization(Athena Scientific, 1998), Introduction to Linear Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998).

Convex Optimization

Stephen P. Boyd,Lieven Vandenberghe
Convex Optimization Book PDF

Author : Stephen P. Boyd,Lieven Vandenberghe
Publisher : Cambridge University Press
Page : 744 pages
File Size : 48,9 Mb
Release : 2004-03-08
Category : Business & Economics
ISBN : 0521833787

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Convex Optimization by Stephen P. Boyd,Lieven Vandenberghe Pdf

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.

Convex Analysis and Nonlinear Optimization

Jonathan Borwein,Adrian S. Lewis
Convex Analysis and Nonlinear Optimization Book PDF

Author : Jonathan Borwein,Adrian S. Lewis
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 49,8 Mb
Release : 2010-05-05
Category : Mathematics
ISBN : 9780387312569

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Convex Analysis and Nonlinear Optimization by Jonathan Borwein,Adrian S. Lewis Pdf

Optimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This book aims to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. Each section concludes with an often extensive set of optional exercises. This new edition adds material on semismooth optimization, as well as several new proofs.

Convex Analysis and Optimization

Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar
Convex Analysis and Optimization Book PDF

Author : Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar
Publisher : Athena Scientific
Page : 560 pages
File Size : 54,9 Mb
Release : 2003-03-01
Category : Mathematics
ISBN : 9781886529458

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Convex Analysis and Optimization by Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar Pdf

A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html

Convex Optimization Algorithms

Dimitri Bertsekas
Convex Optimization Algorithms Book PDF

Author : Dimitri Bertsekas
Publisher : Athena Scientific
Page : 576 pages
File Size : 53,8 Mb
Release : 2015-02-01
Category : Mathematics
ISBN : 9781886529281

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Convex Optimization Algorithms by Dimitri Bertsekas Pdf

This book provides a comprehensive and accessible presentation of algorithms for solving convex optimization problems. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. This is facilitated by the extensive use of analytical and algorithmic concepts of duality, which by nature lend themselves to geometrical interpretation. The book places particular emphasis on modern developments, and their widespread applications in fields such as large-scale resource allocation problems, signal processing, and machine learning. The book is aimed at students, researchers, and practitioners, roughly at the first year graduate level. It is similar in style to the author's 2009"Convex Optimization Theory" book, but can be read independently. The latter book focuses on convexity theory and optimization duality, while the present book focuses on algorithmic issues. The two books share notation, and together cover the entire finite-dimensional convex optimization methodology. To facilitate readability, the statements of definitions and results of the "theory book" are reproduced without proofs in Appendix B.

Convex Optimization

Arto Ruud
Convex Optimization Book PDF

Author : Arto Ruud
Publisher : Nova Science Publishers
Page : 0 pages
File Size : 50,8 Mb
Release : 2019
Category : Convex functions
ISBN : 153614696X

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Convex Optimization by Arto Ruud Pdf

Over the past two decades, it has been recognized that advanced image processing techniques provide valuable information to physicians for the diagnosis, image guided therapy and surgery, and monitoring of human diseases. Convex Optimization: Theory, Methods and Applications introduces novel and sophisticated mathematical problems which encourage the development of advanced optimization and computing methods, especially convex optimization.The authors go on to study Steffensen-King-type methods of convergence to approximate a locally unique solution of a nonlinear equation and also in problems of convex optimization. Real-world applications are also provided.The following study is focused on the design and testing of a Matlab code of the Frank-Wolfe algorithm. The Nesterov step is proposed in order to accelerate the algorithm, and the results of some numerical experiments of constraint optimization are also provided.Lagrangian methods for numerical solutions to constrained convex programs are also explored. For enhanced algorithms, the traditional Lagrange multiplier update is modified to take a soft reflection across the zero boundary. This, coupled with a modified drift expression, is shown to yield improved performance.Next, Newton's mesh independence principle was used to solve a certain class of optimal design problems from earlier studies. Motivated by optimization considerations, the authors show that under the same computational cost, a finer mesh independence principle can be given than before.This compilation closes with a presentation on a local convergence analysis for eighth�order variants of Hansen�Patrick�s family for approximating a locally unique solution of a nonlinear equation. The radius of convergence and computable error bounds on the distances involved are also provided.

Convex Optimization in Normed Spaces

Juan Peypouquet
Convex Optimization in Normed Spaces Book PDF

Author : Juan Peypouquet
Publisher : Springer
Page : 124 pages
File Size : 50,8 Mb
Release : 2015-03-18
Category : Mathematics
ISBN : 9783319137100

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Convex Optimization in Normed Spaces by Juan Peypouquet Pdf

This work is intended to serve as a guide for graduate students and researchers who wish to get acquainted with the main theoretical and practical tools for the numerical minimization of convex functions on Hilbert spaces. Therefore, it contains the main tools that are necessary to conduct independent research on the topic. It is also a concise, easy-to-follow and self-contained textbook, which may be useful for any researcher working on related fields, as well as teachers giving graduate-level courses on the topic. It will contain a thorough revision of the extant literature including both classical and state-of-the-art references.

Lectures on Convex Optimization

Yurii Nesterov
Lectures on Convex Optimization Book PDF

Author : Yurii Nesterov
Publisher : Springer
Page : 589 pages
File Size : 45,8 Mb
Release : 2018-11-19
Category : Mathematics
ISBN : 9783319915784

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Lectures on Convex Optimization by Yurii Nesterov Pdf

This book provides a comprehensive, modern introduction to convex optimization, a field that is becoming increasingly important in applied mathematics, economics and finance, engineering, and computer science, notably in data science and machine learning. Written by a leading expert in the field, this book includes recent advances in the algorithmic theory of convex optimization, naturally complementing the existing literature. It contains a unified and rigorous presentation of the acceleration techniques for minimization schemes of first- and second-order. It provides readers with a full treatment of the smoothing technique, which has tremendously extended the abilities of gradient-type methods. Several powerful approaches in structural optimization, including optimization in relative scale and polynomial-time interior-point methods, are also discussed in detail. Researchers in theoretical optimization as well as professionals working on optimization problems will find this book very useful. It presents many successful examples of how to develop very fast specialized minimization algorithms. Based on the author’s lectures, it can naturally serve as the basis for introductory and advanced courses in convex optimization for students in engineering, economics, computer science and mathematics.

Convex Optimization for Signal Processing and Communications

Chong-Yung Chi,Wei-Chiang Li,Chia-Hsiang Lin
Convex Optimization for Signal Processing and Communications Book PDF

Author : Chong-Yung Chi,Wei-Chiang Li,Chia-Hsiang Lin
Publisher : CRC Press
Page : 294 pages
File Size : 50,5 Mb
Release : 2017-01-24
Category : Technology & Engineering
ISBN : 9781315349800

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Convex Optimization for Signal Processing and Communications by Chong-Yung Chi,Wei-Chiang Li,Chia-Hsiang Lin Pdf

Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications provides fundamental background knowledge of convex optimization, while striking a balance between mathematical theory and applications in signal processing and communications. In addition to comprehensive proofs and perspective interpretations for core convex optimization theory, this book also provides many insightful figures, remarks, illustrative examples, and guided journeys from theory to cutting-edge research explorations, for efficient and in-depth learning, especially for engineering students and professionals. With the powerful convex optimization theory and tools, this book provides you with a new degree of freedom and the capability of solving challenging real-world scientific and engineering problems.

Generalized Convexity and Optimization

Alberto Cambini,Laura Martein
Generalized Convexity and Optimization Book PDF

Author : Alberto Cambini,Laura Martein
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 54,8 Mb
Release : 2008-10-14
Category : Mathematics
ISBN : 9783540708766

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Generalized Convexity and Optimization by Alberto Cambini,Laura Martein Pdf

The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted.

Conjugate Duality in Convex Optimization

Radu Ioan Bot
Conjugate Duality in Convex Optimization Book PDF

Author : Radu Ioan Bot
Publisher : Springer Science & Business Media
Page : 164 pages
File Size : 52,5 Mb
Release : 2009-12-24
Category : Business & Economics
ISBN : 9783642049002

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Conjugate Duality in Convex Optimization by Radu Ioan Bot Pdf

The results presented in this book originate from the last decade research work of the author in the ?eld of duality theory in convex optimization. The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary and suf?cient optimality conditions and, consequently, in generatingdifferent algorithmic approachesfor solving mathematical programming problems. The investigations made in this work prove the importance of the duality theory beyond these aspects and emphasize its strong connections with different topics in convex analysis, nonlinear analysis, functional analysis and in the theory of monotone operators. The ?rst part of the book brings to the attention of the reader the perturbation approach as a fundamental tool for developing the so-called conjugate duality t- ory. The classical Lagrange and Fenchel duality approaches are particular instances of this general concept. More than that, the generalized interior point regularity conditions stated in the past for the two mentioned situations turn out to be p- ticularizations of the ones given in this general setting. In our investigations, the perturbationapproachrepresentsthestartingpointforderivingnewdualityconcepts for several classes of convex optimization problems. Moreover, via this approach, generalized Moreau–Rockafellar formulae are provided and, in connection with them, a new class of regularity conditions, called closedness-type conditions, for both stable strong duality and strong duality is introduced. By stable strong duality we understand the situation in which strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional.

Convex Analysis for Optimization

Jan Brinkhuis
Convex Analysis for Optimization Book PDF

Author : Jan Brinkhuis
Publisher : Springer Nature
Page : 278 pages
File Size : 46,5 Mb
Release : 2020-05-05
Category : Business & Economics
ISBN : 9783030418045

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Convex Analysis for Optimization by Jan Brinkhuis Pdf

This textbook offers graduate students a concise introduction to the classic notions of convex optimization. Written in a highly accessible style and including numerous examples and illustrations, it presents everything readers need to know about convexity and convex optimization. The book introduces a systematic three-step method for doing everything, which can be summarized as "conify, work, deconify". It starts with the concept of convex sets, their primal description, constructions, topological properties and dual description, and then moves on to convex functions and the fundamental principles of convex optimization and their use in the complete analysis of convex optimization problems by means of a systematic four-step method. Lastly, it includes chapters on alternative formulations of optimality conditions and on illustrations of their use. "The author deals with the delicate subjects in a precise yet light-minded spirit... For experts in the field, this book not only offers a unifying view, but also opens a door to new discoveries in convexity and optimization...perfectly suited for classroom teaching." Shuzhong Zhang, Professor of Industrial and Systems Engineering, University of Minnesota

Optimality Conditions in Convex Optimization

Anulekha Dhara,Joydeep Dutta
Optimality Conditions in Convex Optimization Book PDF

Author : Anulekha Dhara,Joydeep Dutta
Publisher : CRC Press
Page : 446 pages
File Size : 45,7 Mb
Release : 2011-10-17
Category : Business & Economics
ISBN : 9781439868225

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Optimality Conditions in Convex Optimization by Anulekha Dhara,Joydeep Dutta Pdf

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.

Convex Analysis and Global Optimization

Hoang Tuy
Convex Analysis and Global Optimization Book PDF

Author : Hoang Tuy
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 49,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475728095

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Convex Analysis and Global Optimization by Hoang Tuy Pdf

Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development. Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.

Statistical Inference Via Convex Optimization

Anatoli Juditsky,Arkadi Nemirovski
Statistical Inference Via Convex Optimization Book PDF

Author : Anatoli Juditsky,Arkadi Nemirovski
Publisher : Princeton University Press
Page : 655 pages
File Size : 47,9 Mb
Release : 2020-04-07
Category : Mathematics
ISBN : 9780691197296

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Statistical Inference Via Convex Optimization by Anatoli Juditsky,Arkadi Nemirovski Pdf

This authoritative book draws on the latest research to explore the interplay of high-dimensional statistics with optimization. Through an accessible analysis of fundamental problems of hypothesis testing and signal recovery, Anatoli Juditsky and Arkadi Nemirovski show how convex optimization theory can be used to devise and analyze near-optimal statistical inferences. Statistical Inference via Convex Optimization is an essential resource for optimization specialists who are new to statistics and its applications, and for data scientists who want to improve their optimization methods. Juditsky and Nemirovski provide the first systematic treatment of the statistical techniques that have arisen from advances in the theory of optimization. They focus on four well-known statistical problems—sparse recovery, hypothesis testing, and recovery from indirect observations of both signals and functions of signals—demonstrating how they can be solved more efficiently as convex optimization problems. The emphasis throughout is on achieving the best possible statistical performance. The construction of inference routines and the quantification of their statistical performance are given by efficient computation rather than by analytical derivation typical of more conventional statistical approaches. In addition to being computation-friendly, the methods described in this book enable practitioners to handle numerous situations too difficult for closed analytical form analysis, such as composite hypothesis testing and signal recovery in inverse problems. Statistical Inference via Convex Optimization features exercises with solutions along with extensive appendixes, making it ideal for use as a graduate text.