Interior Point Techniques in Optimization

Complementarity, Sensitivity and Algorithms

Author: B. Jansen

Publisher: Springer Science & Business Media

ISBN: 1475755619

Category: Mathematics

Page: 280

View: 7318

Operations research and mathematical programming would not be as advanced today without the many advances in interior point methods during the last decade. These methods can now solve very efficiently and robustly large scale linear, nonlinear and combinatorial optimization problems that arise in various practical applications. The main ideas underlying interior point methods have influenced virtually all areas of mathematical programming including: analyzing and solving linear and nonlinear programming problems, sensitivity analysis, complexity analysis, the analysis of Newton's method, decomposition methods, polynomial approximation for combinatorial problems etc. This book covers the implications of interior techniques for the entire field of mathematical programming, bringing together many results in a uniform and coherent way. For the topics mentioned above the book provides theoretical as well as computational results, explains the intuition behind the main ideas, gives examples as well as proofs, and contains an extensive up-to-date bibliography. Audience: The book is intended for students, researchers and practitioners with a background in operations research, mathematics, mathematical programming, or statistics.
Posted in Mathematics

Interior Point Methods of Mathematical Programming

Author: Tamas Terlaky

Publisher: Springer Science & Business Media

ISBN: 9780792342014

Category: Mathematics

Page: 530

View: 7543

One has to make everything as simple as possible but, never more simple. Albert Einstein Discovery consists of seeing what every body has seen and thinking what nobody has thought. Albert S. ent_Gyorgy; The primary goal of this book is to provide an introduction to the theory of Interior Point Methods (IPMs) in Mathematical Programming. At the same time, we try to present a quick overview of the impact of extensions of IPMs on smooth nonlinear optimization and to demonstrate the potential of IPMs for solving difficult practical problems. The Simplex Method has dominated the theory and practice of mathematical pro gramming since 1947 when Dantzig discovered it. In the fifties and sixties several attempts were made to develop alternative solution methods. At that time the prin cipal base of interior point methods was also developed, for example in the work of Frisch (1955), Caroll (1961), Huard (1967), Fiacco and McCormick (1968) and Dikin (1967). In 1972 Klee and Minty made explicit that in the worst case some variants of the simplex method may require an exponential amount of work to solve Linear Programming (LP) problems. This was at the time when complexity theory became a topic of great interest. People started to classify mathematical programming prob lems as efficiently (in polynomial time) solvable and as difficult (NP-hard) problems. For a while it remained open whether LP was solvable in polynomial time or not. The break-through resolution ofthis problem was obtained by Khachijan (1989).
Posted in Mathematics

Complementarity: Applications, Algorithms and Extensions

Author: Michael C. Ferris,Olvi L. Mangasarian,Jong-Shi Pang

Publisher: Springer Science & Business Media

ISBN: 1475732791

Category: Computers

Page: 404

View: 300

This volume presents state-of-the-art complementarity applications, algorithms, extensions and theory in the form of eighteen papers. These at the International Conference on Com invited papers were presented plementarity 99 (ICCP99) held in Madison, Wisconsin during June 9-12, 1999 with support from the National Science Foundation under Grant DMS-9970102. Complementarity is becoming more widely used in a variety of appli cation areas. In this volume, there are papers studying the impact of complementarity in such diverse fields as deregulation of electricity mar kets, engineering mechanics, optimal control and asset pricing. Further more, application of complementarity and optimization ideas to related problems in the burgeoning fields of machine learning and data mining are also covered in a series of three articles. In order to effectively process the complementarity problems that arise in such applications, various algorithmic, theoretical and computational extensions are covered in this volume. Nonsmooth analysis has an im portant role to play in this area as can be seen from articles using these tools to develop Newton and path following methods for constrained nonlinear systems and complementarity problems. Convergence issues are covered in the context of active set methods, global algorithms for pseudomonotone variational inequalities, successive convex relaxation and proximal point algorithms. Theoretical contributions to the connectedness of solution sets and constraint qualifications in the growing area of mathematical programs with equilibrium constraints are also presented. A relaxation approach is given for solving such problems. Finally, computational issues related to preprocessing mixed complementarity problems are addressed.
Posted in Computers

Interior Point Methods for Linear Optimization

Author: Cornelis Roos,Tamás Terlaky,J.-Ph. Vial

Publisher: Springer Science & Business Media

ISBN: 0387263799

Category: Mathematics

Page: 497

View: 7591

The era of interior point methods (IPMs) was initiated by N. Karmarkar’s 1984 paper, which triggered turbulent research and reshaped almost all areas of optimization theory and computational practice. This book offers comprehensive coverage of IPMs. It details the main results of more than a decade of IPM research. Numerous exercises are provided to aid in understanding the material.
Posted in Mathematics

Large-scale Optimization

Problems and Methods

Author: Vladimir Tsurkov

Publisher: Springer Science & Business Media

ISBN: 1475732430

Category: Computers

Page: 312

View: 353

Decomposition methods aim to reduce large-scale problems to simpler problems. This monograph presents selected aspects of the dimension-reduction problem. Exact and approximate aggregations of multidimensional systems are developed and from a known model of input-output balance, aggregation methods are categorized. The issues of loss of accuracy, recovery of original variables (disaggregation), and compatibility conditions are analyzed in detail. The method of iterative aggregation in large-scale problems is studied. For fixed weights, successively simpler aggregated problems are solved and the convergence of their solution to that of the original problem is analyzed. An introduction to block integer programming is considered. Duality theory, which is widely used in continuous block programming, does not work for the integer problem. A survey of alternative methods is presented and special attention is given to combined methods of decomposition. Block problems in which the coupling variables do not enter the binding constraints are studied. These models are worthwhile because they permit a decomposition with respect to primal and dual variables by two-level algorithms instead of three-level algorithms. Audience: This book is addressed to specialists in operations research, optimization, and optimal control.
Posted in Computers

Interior Point Algorithms

Theory and Analysis

Author: Yinyu Ye

Publisher: John Wiley & Sons

ISBN: 1118030958

Category: Mathematics

Page: 440

View: 2488

The first comprehensive review of the theory and practice of one of today's most powerful optimization techniques. The explosive growth of research into and development of interior point algorithms over the past two decades has significantly improved the complexity of linear programming and yielded some of today's most sophisticated computing techniques. This book offers a comprehensive and thorough treatment of the theory, analysis, and implementation of this powerful computational tool. Interior Point Algorithms provides detailed coverage of all basic and advanced aspects of the subject. Beginning with an overview of fundamental mathematical procedures, Professor Yinyu Ye moves swiftly on to in-depth explorations of numerous computational problems and the algorithms that have been developed to solve them. An indispensable text/reference for students and researchers in applied mathematics, computer science, operations research, management science, and engineering, Interior Point Algorithms: * Derives various complexity results for linear and convex programming * Emphasizes interior point geometry and potential theory * Covers state-of-the-art results for extension, implementation, and other cutting-edge computational techniques * Explores the hottest new research topics, including nonlinear programming and nonconvex optimization.
Posted in Mathematics

The Linear Complementarity Problem

Author: Richard W. Cottle,Jong-Shi Pang,Richard E. Stone

Publisher: SIAM

ISBN: 0898716861

Category: Mathematics

Page: 184

View: 9507

A revised edition of the standard reference on the linear complementarity problem.
Posted in Mathematics

Numerical Optimization

Author: Jorge Nocedal,Stephen Wright

Publisher: Springer Science & Business Media

ISBN: 0387400656

Category: Mathematics

Page: 664

View: 2398

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.
Posted in Mathematics

Subject Guide to Books in Print

An Index to the Publishers' Trade List Annual

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 620

Posted in American literature

Lectures on Modern Convex Optimization

Analysis, Algorithms, and Engineering Applications

Author: Aharon Ben-Tal,Arkadi Nemirovski

Publisher: SIAM

ISBN: 0898714915

Category: Technology & Engineering

Page: 488

View: 3538

Here is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.
Posted in Technology & Engineering

Linear Programming with MATLAB

Author: Michael C. Ferris,Olvi L. Mangasarian,Stephen J. Wright

Publisher: SIAM

ISBN: 0898716438

Category: Mathematics

Page: 266

View: 3275

A self-contained introduction to linear programming using MATLAB® software to elucidate the development of algorithms and theory. Exercises are included in each chapter, and additional information is provided in two appendices and an accompanying Web site. Only a basic knowledge of linear algebra and calculus is required.
Posted in Mathematics

Convex Optimization

Author: Stephen P. Boyd,Lieven Vandenberghe

Publisher: Cambridge University Press

ISBN: 9780521833783

Category: Business & Economics

Page: 716

View: 451

A comprehensive introduction to the tools, techniques and applications of convex optimization.
Posted in Business & Economics

High Performance Optimization

Author: Hans Frenk,Kees Roos,Tamas Terlaky,Shuzhong Zhang

Publisher: Springer Science & Business Media

ISBN: 1475732163

Category: Mathematics

Page: 474

View: 3645

For a long time the techniques of solving linear optimization (LP) problems improved only marginally. Fifteen years ago, however, a revolutionary discovery changed everything. A new `golden age' for optimization started, which is continuing up to the current time. What is the cause of the excitement? Techniques of linear programming formed previously an isolated body of knowledge. Then suddenly a tunnel was built linking it with a rich and promising land, part of which was already cultivated, part of which was completely unexplored. These revolutionary new techniques are now applied to solve conic linear problems. This makes it possible to model and solve large classes of essentially nonlinear optimization problems as efficiently as LP problems. This volume gives an overview of the latest developments of such `High Performance Optimization Techniques'. The first part is a thorough treatment of interior point methods for semidefinite programming problems. The second part reviews today's most exciting research topics and results in the area of convex optimization. Audience: This volume is for graduate students and researchers who are interested in modern optimization techniques.
Posted in Mathematics

Interior-point Polynomial Algorithms in Convex Programming

Author: Yurii Nesterov,Arkadii Nemirovskii

Publisher: SIAM

ISBN: 9781611970791

Category: Convex programming

Page: 405

View: 8617

Specialists working in the areas of optimization, mathematical programming, or control theory will find this book invaluable for studying interior-point methods for linear and quadratic programming, polynomial-time methods for nonlinear convex programming, and efficient computational methods for control problems and variational inequalities. A background in linear algebra and mathematical programming is necessary to understand the book. The detailed proofs and lack of "numerical examples" might suggest that the book is of limited value to the reader interested in the practical aspects of convex optimization, but nothing could be further from the truth. An entire chapter is devoted to potential reduction methods precisely because of their great efficiency in practice.
Posted in Convex programming

Applications of Optimization with Xpress-MP

Author: Christelle Guéret,Christian Prins,Marc Sevaux

Publisher: Twayne Publishers

ISBN: 9780954350307

Category: Linear programming

Page: 349

View: 1814

Posted in Linear programming

Introductory Operations Research

Theory and Applications

Author: Harvir Singh Kasana,Krishna Dev Kumar

Publisher: Springer Science & Business Media

ISBN: 3662080117

Category: Business & Economics

Page: 581

View: 9363

Each concept is discussed from the basics and supported by sufficient mathematical background and worked examples. Suitable for individual or group learning, the book offers numerous end-of-chapter problems for study and review.
Posted in Business & Economics

Recent Advances in Optimization

Proceedings of the 8th French-German Conference on Optimization Trier, July 21–26, 1996

Author: Peter Gritzmann,Reiner Horst,Ekkehard Sachs,Rainer Tichatschke

Publisher: Springer Science & Business Media

ISBN: 364259073X

Category: Mathematics

Page: 379

View: 7826

Posted in Mathematics

Linear Programming

Foundations and Extensions

Author: Robert J Vanderbei

Publisher: Springer Science & Business Media

ISBN: 1475756623

Category: Business & Economics

Page: 450

View: 1354

This book provides an introduction to optimization. It details constrained optimization, beginning with a substantial treatment of linear programming and proceeding to convex analysis, network flows, integer programming, quadratic programming, and convex optimization. Coverage underscores the purpose of optimization: to solve practical problems on a computer. C programs that implement the major algorithms and JAVA tools are available online.
Posted in Business & Economics

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

Author: J. E. Dennis, Jr.,Robert B. Schnabel

Publisher: SIAM

ISBN: 9781611971200

Category: Equations

Page: 378

View: 5345

This book has become the standard for a complete, state-of-the-art description of the methods for unconstrained optimization and systems of nonlinear equations. Originally published in 1983, it provides information needed to understand both the theory and the practice of these methods and provides pseudocode for the problems. The algorithms covered are all based on Newton's method or "quasi-Newton" methods, and the heart of the book is the material on computational methods for multidimensional unconstrained optimization and nonlinear equation problems. The republication of this book by SIAM is driven by a continuing demand for specific and sound advice on how to solve real problems. The level of presentation is consistent throughout, with a good mix of examples and theory, making it a valuable text at both the graduate and undergraduate level. It has been praised as excellent for courses with approximately the same name as the book title and would also be useful as a supplemental text for a nonlinear programming or a numerical analysis course. Many exercises are provided to illustrate and develop the ideas in the text. A large appendix provides a mechanism for class projects and a reference for readers who want the details of the algorithms. Practitioners may use this book for self-study and reference. For complete understanding, readers should have a background in calculus and linear algebra. The book does contain background material in multivariable calculus and numerical linear algebra.
Posted in Equations