Computation of the Search Direction in Constrained Optimization Algorithms PDF Download
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Author: Stanford University. Department of Operations Research. Systems Optimization Laboratory
Publisher:
ISBN:
Category :
Languages : en
Pages : 36
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Book Description
Author: Stanford University. Department of Operations Research. Systems Optimization Laboratory
Publisher:
ISBN:
Category :
Languages : en
Pages : 36
Get Book Here
Book Description
Author: Minxin Zhang
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Projected-search methods for bound-constrained optimization are based on searching along a continuous path obtained by projecting a search direction onto the feasible region. These methods have the potential to change the direction of the search path multiple times along the boundary of the feasible region at the cost of computing a single direction. However, as the objective function is only piecewise differentiable along the path, conventional projected-search methods are limited at using a simple backtracking procedure to obtain a step that satisfies an "Armijo-like" sufficient decrease condition. To extend the benefits of Wolfe line search for unconstrained optimization to projected-search methods, a new quasi-Wolfe step is introduced. Two general classes of projected-search methods that use the new quasi-Wolfe search are then formulated and analyzed. These methods may be broadly categorized as either active-set methods or interior methods. Additionally, a new quasi-Newton projected-search method UBOPT is proposed for unconstrained and bound-constrained optimization. The method computes quasi-Newton directions in a sequence of subspaces, and employs the framework of the class of projected-search active-set methods. Furthermore, a new interior method is proposed for general nonlinearly constrained optimization, combining a shifted primal-dual interior method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. The shifts allow the method to be safely "warm started" from a good approximate solution and eliminate the ill-conditioning of the associated linear equations that may occur when the variables are close to zero. The approximate Newton direction is used in conjunction with a new projected-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results demonstrate that the new method requires significantly fewer iterations than a conventional interior method, thereby reducing the number of times that the search direction need be computed.
Author: A. Bachem
Publisher: Springer Science & Business Media
ISBN: 3642688748
Category : Mathematics
Languages : en
Pages : 662
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Book Description
In the late forties, Mathematical Programming became a scientific discipline in its own right. Since then it has experienced a tremendous growth. Beginning with economic and military applications, it is now among the most important fields of applied mathematics with extensive use in engineering, natural sciences, economics, and biological sciences. The lively activity in this area is demonstrated by the fact that as early as 1949 the first "Symposium on Mathe matical Programming" took place in Chicago. Since then mathematical programmers from all over the world have gath ered at the intfrnational symposia of the Mathematical Programming Society roughly every three years to present their recent research, to exchange ideas with their colleagues and to learn about the latest developments in their own and related fields. In 1982, the XI. International Symposium on Mathematical Programming was held at the University of Bonn, W. Germany, from August 23 to 27. It was organized by the Institut fUr Okonometrie und Operations Re search of the University of Bonn in collaboration with the Sonderforschungs bereich 21 of the Deutsche Forschungsgemeinschaft. This volume constitutes part of the outgrowth of this symposium and docu ments its scientific activities. Part I of the book contains information about the symposium, welcoming addresses, lists of committees and sponsors and a brief review about the Ful kerson Prize and the Dantzig Prize which were awarded during the opening ceremony.
Author: Andreas Antoniou
Publisher: Springer Nature
ISBN: 1071608436
Category : Computers
Languages : en
Pages : 737
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Book Description
This textbook provides a hands-on treatment of the subject of optimization. A comprehensive set of problems and exercises makes it suitable for use in one or two semesters of an advanced undergraduate course or a first-year graduate course. Each half of the book contains a full semester’s worth of complementary yet stand-alone material. The practical orientation of the topics chosen and a wealth of useful examples also make the book suitable as a reference work for practitioners in the field. In this second edition the authors have added sections on recent innovations, techniques, and methodologies.
Author: Jorge Nocedal
Publisher: Springer Science & Business Media
ISBN: 0387227423
Category : Mathematics
Languages : en
Pages : 651
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Book Description
The new edition of this book presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on methods best suited to practical problems. This edition has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are widely used in practice and are the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience.
Author: P. Adby
Publisher: Springer Science & Business Media
ISBN: 940095705X
Category : Science
Languages : en
Pages : 214
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Book Description
During the last decade the techniques of non-linear optim ization have emerged as an important subject for study and research. The increasingly widespread application of optim ization has been stimulated by the availability of digital computers, and the necessity of using them in the investigation of large systems. This book is an introduction to non-linear methods of optimization and is suitable for undergraduate and post graduate courses in mathematics, the physical and social sciences, and engineering. The first half of the book covers the basic optimization techniques including linear search methods, steepest descent, least squares, and the Newton-Raphson method. These are described in detail, with worked numerical examples, since they form the basis from which advanced methods are derived. Since 1965 advanced methods of unconstrained and constrained optimization have been developed to utilise the computational power of the digital computer. The second half of the book describes fully important algorithms in current use such as variable metric methods for unconstrained problems and penalty function methods for constrained problems. Recent work, much of which has not yet been widely applied, is reviewed and compared with currently popular techniques under a few generic main headings. vi PREFACE Chapter I describes the optimization problem in mathemat ical form and defines the terminology used in the remainder of the book. Chapter 2 is concerned with single variable optimization. The main algorithms of both search and approximation methods are developed in detail since they are an essential part of many multi-variable methods.
Author: Michael William Ferry
Publisher:
ISBN: 9781124628189
Category :
Languages : en
Pages : 134
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Book Description
Many algorithms used in unconstrained minimization are line-search methods. Given an initial point x and function f : Rn [arrow] R to be minimized, a line-search method repeatedly solves two subproblems : the first calculates a search direction p; the second performs a line search on the function [phi]([alpha]) = f(x + [alpha]p). Then, [alpha]p is added to x and the process is repeated until a solution is located. Quasi-Newton methods are often used to calculate the search direction. A quasi-Newton method creates a quadratic model of f at x and defines the search direction p such that x + p is the minimizer of the model. After each iteration the model is updated to more closely resemble f near x. Line searches seek to satisfy conditions that ensure the convergence of the sequence of iterates. One step that decreases f "sufficiently" is called an Armijo step. A Wolfe step satisfies stronger conditions that impose bounds on [phi]([alpha]). Quasi-Newton methods perform significantly better when using Wolfe steps. Recently Gill and Leonard proposed the reduced Hessian (RH) method, which is a new quasi-Newton method for unconstrained optimization. This method exploits key structures in the quadratic model so that the dimension of the search space is reduced. Placing box constraints x leads to more complex problems. One method for solving such problems is the projected-search method. This method performs an unconstrained minimization on a changing subset of the variables and projects points that violate the constraints back into the feasible region while performing the line search. To date, projected line-search methods have been restricted to using an Armijo-like line search. By modifying the line-search conditions, we create a new projected line search that uses a Wolfe-like step. This line search retains many of the benefits of a Wolfe line search for the unconstrained case. Projected-search methods and RH methods share a similar structure in solving for the search direction. We exploit this similarity and merge the two ideas to create a class of RH methods for box-constrained optimization. When combined with the new line search, this new family of algorithms minimizes problems in less than 74% of the time taken by the leading comparable alternative on a collection of standard test problems.
Author: Shashi Kant Mishra
Publisher: Springer Nature
ISBN: 9811508941
Category : Mathematics
Languages : en
Pages : 309
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Book Description
This book discusses unconstrained optimization with R—a free, open-source computing environment, which works on several platforms, including Windows, Linux, and macOS. The book highlights methods such as the steepest descent method, Newton method, conjugate direction method, conjugate gradient methods, quasi-Newton methods, rank one correction formula, DFP method, BFGS method and their algorithms, convergence analysis, and proofs. Each method is accompanied by worked examples and R scripts. To help readers apply these methods in real-world situations, the book features a set of exercises at the end of each chapter. Primarily intended for graduate students of applied mathematics, operations research and statistics, it is also useful for students of mathematics, engineering, management, economics, and agriculture.
Author: Philip E. Gill
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 312
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Book Description
Author: Michael Bartholomew-Biggs
Publisher: Springer Science & Business Media
ISBN: 0387787232
Category : Mathematics
Languages : en
Pages : 296
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Book Description
This textbook examines a broad range of problems in science and engineering, describing key numerical methods applied to real life. The case studies presented are in such areas as data fitting, vehicle route planning and optimal control, scheduling and resource allocation, sensitivity calculations and worst-case analysis. Chapters are self-contained with exercises provided at the end of most sections. Nonlinear Optimization with Engineering Applications is ideal for self-study and classroom use in engineering courses at the senior undergraduate or graduate level. The book will also appeal to postdocs and advanced researchers interested in the development and use of optimization algorithms.
