NONSTANDARD SIMPLEX ALGORITHM FOR LINEAR

PROGRAMMING

Southeast University Ping-Qi Pan

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.

1. Introduction1.1. 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)

* Affine scaling algorithm ( Dikin 1967)

1.2. Advantages and disadvantages

1.3. Pan’s 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)

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.