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Mathematical Developments Arising from Linear Programming
Edited by: Jeffrey C. Lagarias and Michael J. Todd
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Contemporary Mathematics
1991; 341 pp; softcover
Volume: 114
ISBN-10: 0-8218-5121-7
ISBN-13: 978-0-8218-5121-0
List Price: US$74
Member Price: US$59.20
Order Code: CONM/114
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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.

Table of Contents

Recent progress and new directions
  • C. W. Lee -- Some recent results on convex polytopes
  • K. H. Borgwardt -- Probabilistic analysis of the simplex method
  • N. Megiddo -- On solving the linear programming problem approximately
  • N. Karmarkar -- Riemannian geometry underlying interior-point methods for linear programming
  • A. M. Bloch -- Steepest descent, linear programming, and Hamiltonian flows
Interior-point methods for linear programming
  • Y. Ye -- An \(O(n^3L)\) potential reduction algorithm for linear programming
  • R. J. Vanderbei and J. C. Lagarias -- I. I. Dikin's convergence result for the affine-scaling algorithm
  • I. J. Lustig -- Phase 1 search directions for a primal-dual interior point method for linear programming
  • E. R. Barnes -- Some results concerning convergence of the affine scaling algorithm
  • K. M. Anstreicher -- Dual ellipsoids and degeneracy in the projective algorithm for linear programming
  • M. D. Ašić, V. V. Kovačević-Vujčić, and M. D. Radosavljević-Nikolić -- A note on limiting behavior of the projective and the affine rescaling algorithms
Trajectories of interior-point methods
  • C. Witzgall, P. T. Boggs, and P. D. Domich -- On the convergence behavior of trajectories for linear programming
  • I. Adler and R. D. C. Monteiro -- Limiting behavior of the affine scaling continuous trajectories for linear programming problems
  • R. D. C. Monteiro -- Convergence and boundary behavior of the projective scaling trajectories for linear programming
Nonlinear optimization
  • F. Jarre, G. Sonnevend, and J. Stoer -- On the complexity of a numerical algorithm for solving generalized convex quadratic programs by following a central path
  • B. Kalantari -- Canonical problems for quadratic programming and projective methods for their solution
  • S. Mehrotra and J. Sun -- An interior point algorithm for solving smooth convex programs based on Newton's method
  • A. A. Goldstein -- A modified Kantorovich inequality for the convergence of Newton's method
Integer programming and multi-objective programming
  • N. Karmarkar -- An interior-point approach to NP-complete problems--Part I
  • J. E. Mitchell and M. J. Todd -- Solving matching problems using Karmarkar's algorithm
  • S. S. Abhyankar, T. L. Morin, and T. Trafalis -- Efficient faces of polytopes: Interior point algorithms, parameterization of algebraic varieties, and multiple objective optimization
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