I= POLYHEDRAL COMPUTATIONS OF LINEAR PROGRAMMING
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Title of Thesis
POLYHEDRAL COMPUTATIONS OF LINEAR PROGRAMMING

Author(s)
ALI DINO
Institute/University/Department Details
Department of Mathematics/ University of Karachi
Session
2007
Subject
Mathematics
Number of Pages
156
Keywords (Extracted from title, table of contents and abstract of thesis)
polyhedral computations, linear programming, linear inequality constraints, linear function, hyperplanes, convex polytopes

Abstract
A linear programming, LP, problem is to find a maximum or minimum of a linear function subject to linear inequality constraints. Theoretically every rational LP is solvable in polynomial time by both the ellipsoid method of Khachian and various interior point methods of Karmarker. In practice, very large LPā€™s can be solved efficiently by both the Simplex method and Interior-Point methods.

In 1957, W. M. Hirsch, made a conjecture which says that the diameter of a convex polytope P is bounded by m - n, which imposes bounds on solving and LP problem. That is to say, Simplex Method has although never observed on practical problems, the poor worst-case behavior of the algorithm---the number of iterations may be exponential in the number of unknowns --- led to an ongoing search for algorithms with better computational complexity.

A few researchers are still looking for the refinement of the simplex method that is polynomial time, the combinatorial method for solving linear programming problems and we try in this thesis to contribute towards a newer refinement.

Download Full Thesis
1250.65 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
60.68 KB
2 1 Introduction
135.87 KB
3 2 Geometry of Linear Programs
192.28 KB
  2.1 Point Sets
  2.2 Lines and Hyperplanes
  2.3 Convex Sets
  2.4 Convex Polytopes
4 3 Graphs and Maps
284.68 KB
  3.1 Traversability
  3.2 Partitioning of Graph
5 4 Separation and Decomposition
92.94 KB
  4.1 Diameter
6 5 Polyhedral Computations
348.59 KB
  5.1 Simplex
  5.2 Cross Polytopes
  5.3 Parallelotope & Measure Polytope
  5.4 Tverbergā€™Theorem
  5.5 Radon
  5.6 Helly
  5.7 Caratheodory's Fundamental Theorems
  5.8 Minkoweski - Weyl Theorem
  5.9 Convex Hull
  5.10 Vertex Enumeration
  5.11 Voronoi and Delaunay Triangulations
7 6 Projective Invariants
226.32 KB
  6.1 Desargues and Pappus
  6.2 Perspectivities and Projectivities
8 7 Conclusion
60.27 KB
9 8 Appendix
44.24 KB
  8.1 Appendix I
  8.2 Appendix II
  8.3 Appendix III
  8.4 Appendix IV
  8.5 Appendix V
  8.6 Appendix VI
  8.7 Reference