Mathematical Developments Arising from Linear Programming

Proceedings of a Joint Summer Research Conference Held at Bowdoin College, June 25-July 1, 1988


Author: Jeffrey C. Lagarias,Michael J. Todd

Publisher: American Mathematical Soc.

ISBN: 0821851217

Category: Mathematics

Page: 341

View: 3565

In recent years, there has been intense work in linear and nonlinear programming, much of it centered on understanding and extending the ideas underlying N. Karmarkar's interior-point linear programming algorithm, which was presented in 1984. This interdisciplinary research was the subject of an AMS Summer Research Conference on Mathematical Developments Arising from Linear Programming, held at Bowdoin College in the summer of 1988, which brought together researchers in mathematics, computer science, and operations research. This volume contains the proceedings from the conference. Among the topics covered in this book are: completely integrable dynamical systems arising in optimization problems, Riemannian geometry and interior-point linear programming methods, concepts of approximate solution of linear programs, average case analysis of the simplex method, and recent results in convex polytopes. Some of the papers extend interior-point methods to quadratic programming, the linear complementarity problem, convex programming, multi-criteria optimization, and integer programming. Other papers study the continuous trajectories underlying interior point methods. This book will be an excellent resource for those interested in the latest developments arising from Karmarkar's linear programming algorithm and in path-following methods for solving differential equations.

Acta numerica


Author: [Anonymus AC00403122],[Anonymus AC03502176]

Publisher: Cambridge University Press

ISBN: 9780521410267


Page: N.A

View: 4774

Presents information on "Acta Numerica," an annual publication from Cambridge University Press. Lists the members of the editorial board. Notes that the journal is a collection of review articles, including survey papers by leading researchers in numerical analysis and scientific computing. Offers instructions for contributors and the tables of contents for the current and previous issues.


Proceedings of the Second International Conference on Industrial and Applied Mathematics


Author: Robert E. O'Malley

Publisher: SIAM

ISBN: 9780898713022

Category: Mathematics

Page: 391

View: 9493

Proceedings -- Computer Arithmetic, Algebra, OOP.

Linear Programming 2

Theory and Extensions


Author: George B. Dantzig,Mukund N. Thapa

Publisher: Springer Science & Business Media

ISBN: 0387215697

Category: Mathematics

Page: 448

View: 3475

George Dantzig is widely regarded as the founder of this subject with his invention of the simplex algorithm in the 1940's. In this second volume, the theory of the items discussed in the first volume is expanded to include such additional advanced topics as variants of the simplex method; interior point methods, GUB, decomposition, integer programming, and game theory. Graduate students in the fields of operations research, industrial engineering and applied mathematics will thus find this volume of particular interest.

Linear Programming

A Modern Integrated Analysis


Author: Romesh Saigal

Publisher: Springer Science & Business Media

ISBN: 1461523117

Category: Business & Economics

Page: 342

View: 8787

In Linear Programming: A Modern Integrated Analysis, both boundary (simplex) and interior point methods are derived from the complementary slackness theorem and, unlike most books, the duality theorem is derived from Farkas's Lemma, which is proved as a convex separation theorem. The tedium of the simplex method is thus avoided. A new and inductive proof of Kantorovich's Theorem is offered, related to the convergence of Newton's method. Of the boundary methods, the book presents the (revised) primal and the dual simplex methods. An extensive discussion is given of the primal, dual and primal-dual affine scaling methods. In addition, the proof of the convergence under degeneracy, a bounded variable variant, and a super-linearly convergent variant of the primal affine scaling method are covered in one chapter. Polynomial barrier or path-following homotopy methods, and the projective transformation method are also covered in the interior point chapter. Besides the popular sparse Cholesky factorization and the conjugate gradient method, new methods are presented in a separate chapter on implementation. These methods use LQ factorization and iterative techniques.

Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces


Author: Silvestru Sever Dragomir

Publisher: Springer Science & Business Media

ISBN: 331901448X

Category: Mathematics

Page: 120

View: 627

Aimed toward researchers, postgraduate students, and scientists in linear operator theory and mathematical inequalities, this self-contained monograph focuses on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. Students at the graduate level will learn some essentials that may be useful for reference in courses in functional analysis, operator theory, differential equations, and quantum computation, to name several. Chapter 1 presents fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces. Chapter 2 illustrates recent results obtained concerning numerical radius and norm inequalities for one operator on a complex Hilbert space, as well as some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ryff and Clarke as well as some reverse Schwarz inequalities and Grüss type inequalities obtained by the author. Chapter 3 presents recent results regarding the norms and the numerical radii of two bounded linear operators. The techniques shown in this chapter are elementary but elegant and may be accessible to undergraduate students with a working knowledge of operator theory. A number of vector inequalities in inner product spaces as well as inequalities for means of nonnegative real numbers are also employed in this chapter. All the results presented are completely proved and the original references are mentioned.

Interior Point Approach to Linear, Quadratic, and Convex Programming

Algorithms and Complexity


Author: D. Den Hertog

Publisher: Springer


Category: Mathematics

Page: 208

View: 2331

In this book the rapidly developing field of Interior Point Methods (IPMs) is described. An extensive analysis of so-called path-following methods for linear programming, quadratic programming and convex programming is given. 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.

Mathematical Programming

Recent Developments and Applications


Author: Masao Iri,Kunio Tanabe

Publisher: Springer


Category: Computers

Page: 382

View: 4380