› › On The Relationship Between Conjugate Gradient And Optimal First Order Methods For Convex Optimization

First-Order Methods in Optimization

Author : Amir Beck

Publisher : SIAM

Page : 476 pages

File Size : 33,65 MB

Release : 2017-10-02

Category : Mathematics

ISBN : 1611974984

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Book Description

The primary goal of this book is to provide a self-contained, comprehensive study of the main ?rst-order methods that are frequently used in solving large-scale problems. First-order methods exploit information on values and gradients/subgradients (but not Hessians) of the functions composing the model under consideration. With the increase in the number of applications that can be modeled as large or even huge-scale optimization problems, there has been a revived interest in using simple methods that require low iteration cost as well as low memory storage. The author has gathered, reorganized, and synthesized (in a unified manner) many results that are currently scattered throughout the literature, many of which cannot be typically found in optimization books. First-Order Methods in Optimization offers comprehensive study of first-order methods with the theoretical foundations; provides plentiful examples and illustrations; emphasizes rates of convergence and complexity analysis of the main first-order methods used to solve large-scale problems; and covers both variables and functional decomposition methods.



Nonlinear Conjugate Gradient Methods for Unconstrained Optimization

Author : Neculai Andrei

Publisher : Springer Nature

Page : 515 pages

File Size : 22,47 MB

Release : 2020-06-23

Category : Mathematics

ISBN : 3030429504

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Book Description

Two approaches are known for solving large-scale unconstrained optimization problems—the limited-memory quasi-Newton method (truncated Newton method) and the conjugate gradient method. This is the first book to detail conjugate gradient methods, showing their properties and convergence characteristics as well as their performance in solving large-scale unconstrained optimization problems and applications. Comparisons to the limited-memory and truncated Newton methods are also discussed. Topics studied in detail include: linear conjugate gradient methods, standard conjugate gradient methods, acceleration of conjugate gradient methods, hybrid, modifications of the standard scheme, memoryless BFGS preconditioned, and three-term. Other conjugate gradient methods with clustering the eigenvalues or with the minimization of the condition number of the iteration matrix, are also treated. For each method, the convergence analysis, the computational performances and the comparisons versus other conjugate gradient methods are given. The theory behind the conjugate gradient algorithms presented as a methodology is developed with a clear, rigorous, and friendly exposition; the reader will gain an understanding of their properties and their convergence and will learn to develop and prove the convergence of his/her own methods. Numerous numerical studies are supplied with comparisons and comments on the behavior of conjugate gradient algorithms for solving a collection of 800 unconstrained optimization problems of different structures and complexities with the number of variables in the range [1000,10000]. The book is addressed to all those interested in developing and using new advanced techniques for solving unconstrained optimization complex problems. Mathematical programming researchers, theoreticians and practitioners in operations research, practitioners in engineering and industry researchers, as well as graduate students in mathematics, Ph.D. and master students in mathematical programming, will find plenty of information and practical applications for solving large-scale unconstrained optimization problems and applications by conjugate gradient methods.



Acceleration Methods

Author : Alexandre d'Aspremont

Publisher :

Page : 262 pages

File Size : 27,25 MB

Release : 2021-12-15

Category : Technology & Engineering

ISBN : 9781680839289

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Book Description

This monograph covers recent advances in a range of acceleration techniques frequently used in convex optimization. Using quadratic optimization problems, the authors introduce two key families of methods, namely momentum and nested optimization schemes. These methods are covered in detail and include Chebyshev Acceleration, Nonlinear Acceleration, Nesterov Acceleration, Proximal Acceleration and Catalysts and Restart Schemes.This book provides the reader with an in-depth description of the developments in Acceleration Methods since the early 2000s, whilst referring the reader back to underpinning earlier work for further understanding. This topic is important in the modern-day application of convex optimization techniques in many applicable areas.This book is an introduction to the topic that enables the reader to quickly understand the important principles and apply the techniques to their own research.



Convex Analysis and Monotone Operator Theory in Hilbert Spaces

Author : Heinz H. Bauschke

Publisher : Springer

Page : 624 pages

File Size : 16,93 MB

Release : 2017-02-28

Category : Mathematics

ISBN : 3319483110

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Book Description

This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces. The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, machine learning, physics, decision sciences, economics, and inverse problems. The second edition of Convex Analysis and Monotone Operator Theory in Hilbert Spaces greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated. Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada. Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.



Convex Optimization

Author : Sébastien Bubeck

Publisher : Foundations and Trends (R) in Machine Learning

Page : 142 pages

File Size : 11,40 MB

Release : 2015-11-12

Category : Convex domains

ISBN : 9781601988607

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Book Description

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. It begins with the fundamental theory of black-box optimization and proceeds to guide the reader through recent advances in structural optimization and stochastic optimization. The presentation of black-box optimization, strongly influenced by the seminal book by Nesterov, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. Special attention is also given to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging), and discussing their relevance in machine learning. The text provides a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization it discusses stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. It also briefly touches upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.



Proximal Algorithms

Author : Neal Parikh

Publisher : Now Pub

Page : 130 pages

File Size : 30,45 MB

Release : 2013-11

Category : Mathematics

ISBN : 9781601987167

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Book Description

Proximal Algorithms discusses proximal operators and proximal algorithms, and illustrates their applicability to standard and distributed convex optimization in general and many applications of recent interest in particular. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Proximal Algorithms discusses different interpretations of proximal operators and algorithms, looks at their connections to many other topics in optimization and applied mathematics, surveys some popular algorithms, and provides a large number of examples of proximal operators that commonly arise in practice.



Convex Optimization

Author : Stephen P. Boyd

Publisher : Cambridge University Press

Page : 744 pages

File Size : 14,88 MB

Release : 2004-03-08

Category : Business & Economics

ISBN : 9780521833783

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Book Description

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.



Prediction, Learning, and Games

Author : Nicolo Cesa-Bianchi

Publisher : Cambridge University Press

Page : 4 pages

File Size : 40,4 MB

Release : 2006-03-13

Category : Computers

ISBN : 113945482X

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Book Description

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.



Optimization Methods

Author : H. Tolle

Publisher : Springer Science & Business Media

Page : 239 pages

File Size : 15,82 MB

Release : 2012-12-06

Category : Mathematics

ISBN : 3642877311

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Book Description

Variational problems which are interesting from physical and technical viewpoints are often supplemented with ordinary differential equations as constraints, e. g. , in the form of Newton's equations of motion. Since analytical solutions for such problems are possible only in exceptional cases and numerical treat ment of extensive systems of differential equations formerly caused computational difficulties, in the classical calculus of variations these problems have generally been considered only with respect to their theoretical aspects. However, the advent of digital computer installations has enabled us, approximately since 1950, to make more practical use of the formulas provided by the calculus of variations, and also to proceed from relationships which are oriented more numerically than analytically. This has proved very fruitful since there are areas, in particular, in automatic control and space flight technology, where occasionally even relatively small optimization gains are of interest. Further on, if in a problem we have a free function of time which we may choose as advantageously as possible, then determination of the absolutely optimal course of this function appears always advisable, even if it gives only small improve ments or if it leads to technical difficulties, since: i) we must in any case choose some course for free functions; a criterion which gives an optimal course for that is very practical ii) also, when choosing a certain technically advantageous course we mostly want to know to which extent the performance of the system can further be increased by variation of the free function.



Numerical Optimization

Author : Jorge Nocedal

Publisher : Springer Science & Business Media

Page : 686 pages

File Size : 12,50 MB

Release : 2006-12-11

Category : Mathematics

ISBN : 0387400656

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Book Description

Optimization is an important tool used in decision science and for the analysis of physical systems used in engineering. One can trace its roots to the Calculus of Variations and the work of Euler and Lagrange. This natural and reasonable approach to mathematical programming covers numerical methods for finite-dimensional optimization problems. It begins with very simple ideas progressing through more complicated concepts, concentrating on methods for both unconstrained and constrained optimization.