Arc Search Techniques For Interior Point Methods

Author : Yaguang Yang
Publisher : CRC Press
Page : 306 pages
File Size : 45,9 Mb
Release : 2020-11-26
Category : Mathematics
ISBN : 9781000220131

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Arc-Search Techniques for Interior-Point Methods by Yaguang Yang Pdf

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.

Author : Yurii Nesterov,Arkadii Nemirovskii
Publisher : SIAM
Page : 414 pages
File Size : 48,7 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.

Author : Stephen J. Wright
Publisher : SIAM
Page : 318 pages
File Size : 42,6 Mb
Release : 1997-01-01
Category : Technology & Engineering
ISBN : 089871382X

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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.

Author : Panos M. Pardalos and Henry Wolkowicz
Publisher : American Mathematical Soc.
Page : 276 pages
File Size : 45,8 Mb
Release : 2024-06-28
Category : Interior-point methods
ISBN : 0821871250

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Topics in Semidefinite and Interior-Point Methods by Panos M. Pardalos and 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.

Author : Yinyu Ye
Publisher : John Wiley & Sons
Page : 440 pages
File Size : 45,5 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.

Author : Bin Zhou,Junjie Hu,Liansong Xiong,Jian Zhao,Siqi Bu,Jingyang Fang,Hugo Morais,Peng Hou
Publisher : Frontiers Media SA
Page : 329 pages
File Size : 47,7 Mb
Release : 2023-10-18
Category : Technology & Engineering
ISBN : 9782832536063

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Advanced Technologies for Planning and Operation of Prosumer Energy Systems, Volume II by Bin Zhou,Junjie Hu,Liansong Xiong,Jian Zhao,Siqi Bu,Jingyang Fang,Hugo Morais,Peng Hou Pdf

Author : Stephen J. Wright
Publisher : SIAM
Page : 293 pages
File Size : 44,9 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.

Author : Panos M Pardalos,Athanasios Migdalas,Rainer E Burkard
Publisher : World Scientific
Page : 372 pages
File Size : 49,5 Mb
Release : 2002-04-05
Category : Mathematics
ISBN : 9789814489652

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Combinatorial and Global Optimization by Panos M Pardalos,Athanasios Migdalas,Rainer E Burkard Pdf

Combinatorial and global optimization problems appear in a wide range of applications in operations research, engineering, biological science, and computer science. In combinatorial optimization and graph theory, many approaches have been developed that link the discrete universe to the continuous universe through geometric, analytic, and algebraic techniques. Such techniques include global optimization formulations, semidefinite programming, and spectral theory. Recent major successes based on these approaches include interior point algorithms for linear and discrete problems, the celebrated Goemans–Williamson relaxation of the maximum cut problem, and the Du–Hwang solution of the Gilbert–Pollak conjecture. Since integer constraints are equivalent to nonconvex constraints, the fundamental difference between classes of optimization problems is not between discrete and continuous problems but between convex and nonconvex optimization problems. This volume is a selection of refereed papers based on talks presented at a conference on “Combinatorial and Global Optimization” held at Crete, Greece. Contents:A Forest Exterior Point Algorithm for Assignment Problems (H Achatz et al.)Location/Allocation of Queuing Facilities in Continuous Space Using Minsum and Minimax Criteria (J Brimberg et al.)Algorithms for the Consistency Analysis in Scenario Projects (R Feldmann et al.)Solving Quadratic Knapsack Problems by Reformulation and Tabu Search. Single Constraint Case (F Glover et al.)Global Optimization Using Dynamic Search Trajectories (A A Groenwold & J A Snyman)On Pareto Efficiency. A General Constructive Existence Principle (G Isac)Piecewise Linear Network Flow Problems (D Kim & P M Pardalos)Semidefinite Programming Approaches for MAX-2-SAT and MAX-3-SAT: Computational Perspectives (E de Klerk & J P Warners)Heuristic Solutions of Vehicle Routing Problems in Supply Chain Management (Y Marinakis & A Migdalas)A New Finite Cone Covering Algorithm for Concave Minimization (C Meyer & B Jaumard)Frequency Assignment for Very Large, Sparse Networks (R Murphey)GPS Network Design: An Application of the Simulated Annealing Heuristic Technique (H A Saleh & P J Dare)Normal Branch and Bound Algorithms for General Nonconvex Quadratic Programming (H Tuy)and other papers Readership: Researchers in numerical & computational mathematics, optimization, combinatorics & graph theory, networking and materials engineering. Keywords:Combinatorial Optimization;Global Optimization

Author : Masakazu Kojima,N. Megiddo,T. Noma,Akiko Yoshise
Publisher : Springer Science & Business Media
Page : 124 pages
File Size : 45,5 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.

Author : Jorge Nocedal,Stephen Wright
Publisher : Springer Science & Business Media
Page : 686 pages
File Size : 49,7 Mb
Release : 2006-12-11
Category : Mathematics
ISBN : 9780387400655

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Numerical Optimization by Jorge Nocedal,Stephen Wright Pdf

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.

Author : Nimrod Megiddo
Publisher : Springer Science & Business Media
Page : 164 pages
File Size : 44,6 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."

Author : Ding-Zhu Du,Jie Sun
Publisher : Springer Science & Business Media
Page : 402 pages
File Size : 41,7 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461336297

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Advances in Optimization and Approximation by Ding-Zhu Du,Jie Sun Pdf

This book is a collection of research papers in optimization and approximation dedicated to Professor Minyi Yue of the Institute of Applied Mathematics, Beijing, China. The papers provide a broad spectrum of research on optimization problems, including scheduling, location, assignment, linear and nonlinear programming problems as well as problems in molecular biology. The emphasis of the book is on algorithmic aspects of research work in optimization. Special attention is paid to approximation algorithms, including heuristics for combinatorial approximation problems, approximation algorithms for global optimization problems, and applications of approximations in real problems. The work provides the state of the art for researchers in mathematical programming, operations research, theoretical computer science and applied mathematics.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 49,6 Mb
Release : 2024-06-28
Category : Electronic
ISBN : 9789814477222

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Optimization of Structural and Mechanical Systems by Anonim Pdf

Author : Aharon Ben-Tal,Arkadi Nemirovski
Publisher : SIAM
Page : 500 pages
File Size : 42,5 Mb
Release : 2001-01-01
Category : Technology & Engineering
ISBN : 9780898714913

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Lectures on Modern Convex Optimization by Aharon Ben-Tal,Arkadi Nemirovski 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.

Author : Lap Chi Lau,R. Ravi,Mohit Singh
Publisher : Cambridge University Press
Page : 255 pages
File Size : 48,8 Mb
Release : 2011-04-18
Category : Computers
ISBN : 9781139499392

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Iterative Methods in Combinatorial Optimization by Lap Chi Lau,R. Ravi,Mohit Singh Pdf

With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.