Interior Point Methods Of Mathematical Programming

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Interior Point Methods of Mathematical Programming

Tamás Terlaky
Interior Point Methods of Mathematical Programming Book PDF

Author : Tamás Terlaky
Publisher : Springer Science & Business Media
Page : 544 pages
File Size : 52,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461334491

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Interior Point Methods of Mathematical Programming by Tamás Terlaky Pdf

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

Stephen J. Wright
Primal Dual Interior Point Methods Book PDF

Author : Stephen J. Wright
Publisher : SIAM
Page : 293 pages
File Size : 52,9 Mb
Release : 1997-01-01
Category : Technology & Engineering
ISBN : 9780898713824

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Primal-Dual Interior-Point Methods by Stephen J. Wright Pdf

Presents the major primal-dual algorithms for linear programming. A thorough, straightforward description of the theoretical properties of these methods.

Interior Point Approach to Linear, Quadratic and Convex Programming

D. den Hertog
Interior Point Approach to Linear Quadratic and Convex Programming Book PDF

Author : D. den Hertog
Publisher : Springer Science & Business Media
Page : 210 pages
File Size : 50,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401111348

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Interior Point Approach to Linear, Quadratic and Convex Programming by D. den Hertog Pdf

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 Methods for Linear Optimization

Cornelis Roos,Tamás Terlaky,J.-Ph. Vial
Interior Point Methods for Linear Optimization Book PDF

Author : Cornelis Roos,Tamás Terlaky,J.-Ph. Vial
Publisher : Springer Science & Business Media
Page : 501 pages
File Size : 47,6 Mb
Release : 2006-02-08
Category : Mathematics
ISBN : 9780387263793

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Interior Point Methods for Linear Optimization by Cornelis Roos,Tamás Terlaky,J.-Ph. Vial Pdf

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.

Progress in Mathematical Programming

Nimrod Megiddo
Progress in Mathematical Programming Book PDF

Author : Nimrod Megiddo
Publisher : Springer Science & Business Media
Page : 164 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461396178

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Progress in Mathematical Programming by Nimrod Megiddo Pdf

The starting point of this volume was a conference entitled "Progress in Mathematical Programming," held at the Asilomar Conference Center in Pacific Grove, California, March 1-4, 1987. The main topic of the conference was developments in the theory and practice of linear programming since Karmarkar's algorithm. There were thirty presentations and approximately fifty people attended. Presentations included new algorithms, new analyses of algorithms, reports on computational experience, and some other topics related to the practice of mathematical programming. Interestingly, most of the progress reported at the conference was on the theoretical side. Several new polynomial algorithms for linear program ming were presented (Barnes-Chopra-Jensen, Goldfarb-Mehrotra, Gonzaga, Kojima-Mizuno-Yoshise, Renegar, Todd, Vaidya, and Ye). Other algorithms presented were by Betke-Gritzmann, Blum, Gill-Murray-Saunders-Wright, Nazareth, Vial, and Zikan-Cottle. Efforts in the theoretical analysis of algo rithms were also reported (Anstreicher, Bayer-Lagarias, Imai, Lagarias, Megiddo-Shub, Lagarias, Smale, and Vanderbei). Computational experiences were reported by Lustig, Tomlin, Todd, Tone, Ye, and Zikan-Cottle. Of special interest, although not in the main direction discussed at the conference, was the report by Rinaldi on the practical solution of some large traveling salesman problems. At the time of the conference, it was still not clear whether the new algorithms developed since Karmarkar's algorithm would replace the simplex method in practice. Alan Hoffman presented results on conditions under which linear programming problems can be solved by greedy algorithms."

Interior Point Techniques in Optimization

B. Jansen
Interior Point Techniques in Optimization Book PDF

Author : B. Jansen
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 42,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475755619

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Interior Point Techniques in Optimization by B. Jansen Pdf

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.

Interior-point Polynomial Algorithms in Convex Programming

Yurii Nesterov,Arkadii Nemirovskii
Interior point Polynomial Algorithms in Convex Programming Book PDF

Author : Yurii Nesterov,Arkadii Nemirovskii
Publisher : SIAM
Page : 414 pages
File Size : 45,6 Mb
Release : 1994-01-01
Category : Mathematics
ISBN : 1611970792

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Interior-point Polynomial Algorithms in Convex Programming by Yurii Nesterov,Arkadii Nemirovskii Pdf

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.

Self-Regularity

Jiming Peng,Cornelis Roos,Tamás Terlaky
Self Regularity Book PDF

Author : Jiming Peng,Cornelis Roos,Tamás Terlaky
Publisher : Princeton University Press
Page : 208 pages
File Size : 48,9 Mb
Release : 2009-01-10
Category : Mathematics
ISBN : 9781400825134

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Self-Regularity by Jiming Peng,Cornelis Roos,Tamás Terlaky Pdf

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.

A Mathematical View of Interior-point Methods in Convex Optimization

James Renegar
A Mathematical View of Interior point Methods in Convex Optimization Book PDF

Author : James Renegar
Publisher : SIAM
Page : 124 pages
File Size : 43,6 Mb
Release : 2001-01-01
Category : Mathematics
ISBN : 0898718813

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A Mathematical View of Interior-point Methods in Convex Optimization by James Renegar Pdf

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.

A Mathematical View of Interior-Point Methods in Convex Optimization

James Renegar
A Mathematical View of Interior Point Methods in Convex Optimization Book PDF

Author : James Renegar
Publisher : SIAM
Page : 122 pages
File Size : 49,8 Mb
Release : 2001-01-01
Category : Mathematics
ISBN : 9780898715026

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A Mathematical View of Interior-Point Methods in Convex Optimization by James Renegar Pdf

Takes the reader who knows little of interior-point methods to within sight of the research frontier.

Topics in Semidefinite and Interior-Point Methods

Panos M. Pardalos,Henry Wolkowicz
Topics in Semidefinite and Interior Point Methods Book PDF

Author : Panos M. Pardalos,Henry Wolkowicz
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 55,8 Mb
Release : 1998
Category : Interior-point methods
ISBN : 9780821808252

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Topics in Semidefinite and Interior-Point Methods by Panos M. Pardalos,Henry Wolkowicz Pdf

This volume presents refereed papers presented at the workshop Semidefinite Programming and Interior-Point Approaches for Combinatorial Problems: held at The Fields Institute in May 1996. Semidefinite programming (SDP) is a generalization of linear programming (LP) in that the non-negativity constraints on the variables is replaced by a positive semidefinite constraint on matrix variables. Many of the elegant theoretical properties and powerful solution techniques follow through from LP to SDP. In particular, the primal-dual interior-point methods, which are currently so successful for LP, can be used to efficiently solve SDP problems. In addition to the theoretical and algorithmic questions, SDP has found many important applications in combinatorial optimization, control theory and other areas of mathematical programming. The papers in this volume cover a wide spectrum of recent developments in SDP. The volume would be suitable as a textbook for advanced courses in optimization. It is intended for graduate students and researchers in mathematics, computer science, engineering and operations.

A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems

Masakazu Kojima,N. Megiddo,T. Noma,Akiko Yoshise
A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems Book PDF

Author : Masakazu Kojima,N. Megiddo,T. Noma,Akiko Yoshise
Publisher : Springer Science & Business Media
Page : 124 pages
File Size : 46,6 Mb
Release : 1991-09-25
Category : Language Arts & Disciplines
ISBN : 3540545093

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A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems by Masakazu Kojima,N. Megiddo,T. Noma,Akiko Yoshise Pdf

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.

Interior Point Algorithms

Yinyu Ye
Interior Point Algorithms Book PDF

Author : Yinyu Ye
Publisher : John Wiley & Sons
Page : 440 pages
File Size : 52,6 Mb
Release : 2011-10-11
Category : Mathematics
ISBN : 9781118030950

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Interior Point Algorithms by Yinyu Ye Pdf

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.

Aspects of Semidefinite Programming

E. de Klerk
Aspects of Semidefinite Programming Book PDF

Author : E. de Klerk
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 50,5 Mb
Release : 2006-04-18
Category : Computers
ISBN : 9780306478192

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Aspects of Semidefinite Programming by E. de Klerk Pdf

Semidefinite programming has been described as linear programming for the year 2000. It is an exciting new branch of mathematical programming, due to important applications in control theory, combinatorial optimization and other fields. Moreover, the successful interior point algorithms for linear programming can be extended to semidefinite programming. In this monograph the basic theory of interior point algorithms is explained. This includes the latest results on the properties of the central path as well as the analysis of the most important classes of algorithms. Several "classic" applications of semidefinite programming are also described in detail. These include the Lovász theta function and the MAX-CUT approximation algorithm by Goemans and Williamson. Audience: Researchers or graduate students in optimization or related fields, who wish to learn more about the theory and applications of semidefinite programming.

Theory and Algorithms for Linear Optimization

Cornelis Roos,T. Terlaky,J.-Ph. Vial
Theory and Algorithms for Linear Optimization Book PDF

Author : Cornelis Roos,T. Terlaky,J.-Ph. Vial
Publisher : Unknown
Page : 520 pages
File Size : 45,5 Mb
Release : 1997-03-04
Category : Mathematics
ISBN : STANFORD:36105019761993

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Theory and Algorithms for Linear Optimization by Cornelis Roos,T. Terlaky,J.-Ph. Vial Pdf

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.