NONSTANDARD SIMPLEX ALGORITHM FOR LINEAR PROGRAMMING

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NONSTANDARD SIMPLEX ALGORITHM FOR LINEAR
PROGRAMMING

Southeast University Ping-Qi
Pan
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• Abstract
• The simplex algorithm travels, on the
  underlying
• polyhedron, from vertex to vertex until reaching
  an
• optimal vertex. With the same simplex framework,
  
• the proposed algorithm generates a series of
• feasible points (not necessarily vertices). In
  particular,
• it is exactly an interior point algorithm if the
  initial
• point used is interior. Computational
  experiments
• show that the algorithm is very efficient,
  relative to
• the conventional simplex algorithm. It
  terminates at
• an approximate optimal vertex, or at an optimal
• vertex if a simple purification is incorporated.

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• Introduction
• 1.1. Historical story
• Simplex algorithm (G. B. Dantzig 1947)
• Software (Orchard-Hays 1954)
• Exponential complexity of SA
• (Klee and Minty 1972)
• Degeneracy and cycling
• (Hoffman 1953 and Beale 1955)
• Ellipsoid algorithm ( Khachiyan 1979)
• Interior-point algorithm ( Karmarkar 1984)

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• Affine scaling algorithm ( Dikin 1967)
• 1.2. Advantages and disadvantages
• 1.3. Pans work
• The obtuse-angle principle(1990)
• Bisection simplex algorithm (1991)
• The most-obtuse-angle rules(1997)
• Deficient basis and projective pivot algorithms
  (1997-)
• Nested pricing (2008-)
• Affine-scaling pivot algorithm (submitted)

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• 1.4. Main features of the proposed algorithm
• based on the simplex framework
• uses a new column rule
• can start with any feasible point
• Just an interior-point algorithm if starting with
  an interior point
• achieves an approximate optimal vertex
• achieves an optimal vertex if a simple
  purification is carried out.

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