A Unified Approach To Interior Point Algorithms For Linear Complementarity Problems

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A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems

Author : Masakazu Kojima
Publisher : Springer Science & Business Media
ISBN 13 : 9783540545095
Total Pages : 124 pages
Book Rating : 4.5/5 (45 download)

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Book Synopsis A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems by : Masakazu Kojima

Download or read book A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems written by Masakazu Kojima and published by Springer Science & Business Media. This book was released on 1991-09-25 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: Following Karmarkar's 1984 linear programming algorithm, numerous interior-point algorithms have been proposed for various mathematical programming problems such as linear programming, convex quadratic programming and convex programming in general. This monograph presents a study of interior-point algorithms for the linear complementarity problem (LCP) which is known as a mathematical model for primal-dual pairs of linear programs and convex quadratic programs. A large family of potential reduction algorithms is presented in a unified way for the class of LCPs where the underlying matrix has nonnegative principal minors (P0-matrix). This class includes various important subclasses such as positive semi-definite matrices, P-matrices, P*-matrices introduced in this monograph, and column sufficient matrices. The family contains not only the usual potential reduction algorithms but also path following algorithms and a damped Newton method for the LCP. The main topics are global convergence, global linear convergence, and the polynomial-time convergence of potential reduction algorithms included in the family.

A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems

Author : Masakazu Kojima
Publisher :
ISBN 13 : 9783662207840
Total Pages : 122 pages
Book Rating : 4.2/5 (78 download)

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Book Synopsis A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems by : Masakazu Kojima

Download or read book A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems written by Masakazu Kojima and published by . This book was released on 2014-01-15 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Interior Point Methods of Mathematical Programming

Author : Tamás Terlaky
Publisher : Springer Science & Business Media
ISBN 13 : 1461334497
Total Pages : 544 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Interior Point Methods of Mathematical Programming by : Tamás Terlaky

Download or read book Interior Point Methods of Mathematical Programming written by Tamás Terlaky and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).

Primal-dual Interior-Point Methods

Author : Stephen J. Wright
Publisher : SIAM
ISBN 13 : 9781611971453
Total Pages : 309 pages
Book Rating : 4.9/5 (714 download)

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Book Synopsis Primal-dual Interior-Point Methods by : Stephen J. Wright

Download or read book Primal-dual Interior-Point Methods written by Stephen J. Wright and published by SIAM. This book was released on 1997-01-01 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the past decade, primal-dual algorithms have emerged as the most important and useful algorithms from the interior-point class. This book presents the major primal-dual algorithms for linear programming in straightforward terms. A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software. This is an excellent, timely, and well-written work. The major primal-dual algorithms covered in this book are path-following algorithms (short- and long-step, predictor-corrector), potential-reduction algorithms, and infeasible-interior-point algorithms. A unified treatment of superlinear convergence, finite termination, and detection of infeasible problems is presented. Issues relevant to practical implementation are also discussed, including sparse linear algebra and a complete specification of Mehrotra's predictor-corrector algorithm. Also treated are extensions of primal-dual algorithms to more general problems such as monotone complementarity, semidefinite programming, and general convex programming problems.

Interior Point Techniques in Optimization

Author : B. Jansen
Publisher : Springer Science & Business Media
ISBN 13 : 1475755619
Total Pages : 285 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Interior Point Techniques in Optimization by : B. Jansen

Download or read book Interior Point Techniques in Optimization written by B. Jansen and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Self-Regularity

Author : Jiming Peng
Publisher : Princeton University Press
ISBN 13 : 140082513X
Total Pages : 208 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Self-Regularity by : Jiming Peng

Download or read book Self-Regularity written by Jiming Peng and published by Princeton University Press. This book was released on 2009-01-10 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.

Interior-point Polynomial Algorithms in Convex Programming

Author : Yurii Nesterov
Publisher : SIAM
ISBN 13 : 9781611970791
Total Pages : 414 pages
Book Rating : 4.9/5 (77 download)

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Book Synopsis Interior-point Polynomial Algorithms in Convex Programming by : Yurii Nesterov

Download or read book Interior-point Polynomial Algorithms in Convex Programming written by Yurii Nesterov and published by SIAM. This book was released on 1994-01-01 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Interior Point Approach to Linear, Quadratic and Convex Programming

Author : D. den Hertog
Publisher : Springer Science & Business Media
ISBN 13 : 9401111340
Total Pages : 210 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Interior Point Approach to Linear, Quadratic and Convex Programming by : D. den Hertog

Download or read book Interior Point Approach to Linear, Quadratic and Convex Programming written by D. den Hertog and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the rapidly developing field of interior point methods (IPMs). An extensive analysis is given of path-following methods for linear programming, quadratic programming and convex programming. These methods, which form a subclass of interior point methods, follow the central path, which is an analytic curve defined by the problem. Relatively simple and elegant proofs for polynomiality are given. The theory is illustrated using several explicit examples. Moreover, an overview of other classes of IPMs is given. It is shown that all these methods rely on the same notion as the path-following methods: all these methods use the central path implicitly or explicitly as a reference path to go to the optimum. For specialists in IPMs as well as those seeking an introduction to IPMs. The book is accessible to any mathematician with basic mathematical programming knowledge.

Interior Point Algorithms

Author : Yinyu Ye
Publisher : John Wiley & Sons
ISBN 13 : 1118030958
Total Pages : 440 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Interior Point Algorithms by : Yinyu Ye

Download or read book Interior Point Algorithms written by Yinyu Ye and published by John Wiley & Sons. This book was released on 2011-10-11 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first comprehensive review of the theory and practice of one oftoday's most powerful optimization techniques. The explosive growth of research into and development of interiorpoint algorithms over the past two decades has significantlyimproved the complexity of linear programming and yielded some oftoday's most sophisticated computing techniques. This book offers acomprehensive and thorough treatment of the theory, analysis, andimplementation of this powerful computational tool. Interior Point Algorithms provides detailed coverage of all basicand advanced aspects of the subject. Beginning with an overview offundamental mathematical procedures, Professor Yinyu Ye movesswiftly on to in-depth explorations of numerous computationalproblems and the algorithms that have been developed to solve them.An indispensable text/reference for students and researchers inapplied mathematics, computer science, operations research,management science, and engineering, Interior Point Algorithms: * Derives various complexity results for linear and convexprogramming * 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 nonlinearprogramming and nonconvex optimization.

Theory and Algorithms for Linear Optimization

Author : Cornelis Roos
Publisher :
ISBN 13 :
Total Pages : 520 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis Theory and Algorithms for Linear Optimization by : Cornelis Roos

Download or read book Theory and Algorithms for Linear Optimization written by Cornelis Roos and published by . This book was released on 1997-03-04 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: The approach to LO in this book is new in many aspects. In particular the IPM based development of duality theory is surprisingly elegant. The algorithmic parts of the book contain a complete discussion of many algorithmic variants, including predictor-corrector methods, partial updating, higher order methods and sensitivity and parametric analysis.

Interior Point Methods for Linear Optimization

Author : Cornelis Roos
Publisher : Springer Science & Business Media
ISBN 13 : 0387263799
Total Pages : 501 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Interior Point Methods for Linear Optimization by : Cornelis Roos

Download or read book Interior Point Methods for Linear Optimization written by Cornelis Roos and published by Springer Science & Business Media. This book was released on 2006-02-08 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Primal-dual Interior Point Algorithms for Linear Programming and the Linear Complementarity Problem

Author : Kevin Anthony McShane
Publisher :
ISBN 13 :
Total Pages : 324 pages
Book Rating : 4.E/5 ( download)

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Book Synopsis Primal-dual Interior Point Algorithms for Linear Programming and the Linear Complementarity Problem by : Kevin Anthony McShane

Download or read book Primal-dual Interior Point Algorithms for Linear Programming and the Linear Complementarity Problem written by Kevin Anthony McShane and published by . This book was released on 1992 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Arc-Search Techniques for Interior-Point Methods

Author : Yaguang Yang
Publisher : CRC Press
ISBN 13 : 1000220133
Total Pages : 306 pages
Book Rating : 4.0/5 (2 download)

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Book Synopsis Arc-Search Techniques for Interior-Point Methods by : Yaguang Yang

Download or read book Arc-Search Techniques for Interior-Point Methods written by Yaguang Yang and published by CRC Press. This book was released on 2020-11-26 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses an important area of numerical optimization, called interior-point method. This topic has been popular since the 1980s when people gradually realized that all simplex algorithms were not convergent in polynomial time and many interior-point algorithms could be proved to converge in polynomial time. However, for a long time, there was a noticeable gap between theoretical polynomial bounds of the interior-point algorithms and efficiency of these algorithms. Strategies that were important to the computational efficiency became barriers in the proof of good polynomial bounds. The more the strategies were used in algorithms, the worse the polynomial bounds became. To further exacerbate the problem, Mehrotra's predictor-corrector (MPC) algorithm (the most popular and efficient interior-point algorithm until recently) uses all good strategies and fails to prove the convergence. Therefore, MPC does not have polynomiality, a critical issue with the simplex method. This book discusses recent developments that resolves the dilemma. It has three major parts. The first, including Chapters 1, 2, 3, and 4, presents some of the most important algorithms during the development of the interior-point method around the 1990s, most of them are widely known. The main purpose of this part is to explain the dilemma described above by analyzing these algorithms' polynomial bounds and summarizing the computational experience associated with them. The second part, including Chapters 5, 6, 7, and 8, describes how to solve the dilemma step-by-step using arc-search techniques. At the end of this part, a very efficient algorithm with the lowest polynomial bound is presented. The last part, including Chapters 9, 10, 11, and 12, extends arc-search techniques to some more general problems, such as convex quadratic programming, linear complementarity problem, and semi-definite programming.

Primal-Dual Interior-Point Methods

Author : Stephen J. Wright
Publisher : SIAM
ISBN 13 : 089871382X
Total Pages : 293 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Primal-Dual Interior-Point Methods by : Stephen J. Wright

Download or read book Primal-Dual Interior-Point Methods written by Stephen J. Wright and published by SIAM. This book was released on 1997-01-01 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the major primal-dual algorithms for linear programming. A thorough, straightforward description of the theoretical properties of these methods.

Optimization Theory, Decision Making, and Operations Research Applications

Author : Athanasios Migdalas
Publisher : Springer Science & Business Media
ISBN 13 : 1461451345
Total Pages : 364 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Optimization Theory, Decision Making, and Operations Research Applications by : Athanasios Migdalas

Download or read book Optimization Theory, Decision Making, and Operations Research Applications written by Athanasios Migdalas and published by Springer Science & Business Media. This book was released on 2012-11-28 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: These proceedings consist of 30 selected research papers based on results presented at the 10th Balkan Conference & 1st International Symposium on Operational Research (BALCOR 2011) held in Thessaloniki, Greece, September 22-24, 2011. BALCOR is an established biennial conference attended by a large number of faculty, researchers and students from the Balkan countries but also from other European and Mediterranean countries as well. Over the past decade, the BALCOR conference has facilitated the exchange of scientific and technical information on the subject of Operations Research and related fields such as Mathematical Programming, Game Theory, Multiple Criteria Decision Analysis, Information Systems, Data Mining and more, in order to promote international scientific cooperation. The carefully selected and refereed papers present important recent developments and modern applications and will serve as excellent reference for students, researchers and practitioners in these disciplines.

Linear Programming: Mathematics, Theory and Algorithms

Author : M.J. Panik
Publisher : Springer Science & Business Media
ISBN 13 : 1461334349
Total Pages : 502 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Linear Programming: Mathematics, Theory and Algorithms by : M.J. Panik

Download or read book Linear Programming: Mathematics, Theory and Algorithms written by M.J. Panik and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: Linear Programming provides an in-depth look at simplex based as well as the more recent interior point techniques for solving linear programming problems. Starting with a review of the mathematical underpinnings of these approaches, the text provides details of the primal and dual simplex methods with the primal-dual, composite, and steepest edge simplex algorithms. This then is followed by a discussion of interior point techniques, including projective and affine potential reduction, primal and dual affine scaling, and path following algorithms. Also covered is the theory and solution of the linear complementarity problem using both the complementary pivot algorithm and interior point routines. A feature of the book is its early and extensive development and use of duality theory. Audience: The book is written for students in the areas of mathematics, economics, engineering and management science, and professionals who need a sound foundation in the important and dynamic discipline of linear programming.

Approximation, Optimization and Mathematical Economics

Author : Marc Lassonde
Publisher : Springer Science & Business Media
ISBN 13 : 3642575927
Total Pages : 390 pages
Book Rating : 4.6/5 (425 download)

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Book Synopsis Approximation, Optimization and Mathematical Economics by : Marc Lassonde

Download or read book Approximation, Optimization and Mathematical Economics written by Marc Lassonde and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this proceedings volume reflect the current trends in the theory of approximation, optimization and mathematical economics, and include numerous applications. The book will be of interest to researchers and graduate students involved in functional analysis, approximation theory, mathematical programming and optimization, game theory, mathematical finance and economics.