DUAL FACE ALGORITHM FOR LINEAR PROGRAMMING

SOUTHEAST UNIVERSITY

PING-QI PAN

Abstract

The proposed algorithm proceeds from dual face to dual face, until reaching a dual optimal face along with a pair of dual and primal optimal solutions, compared with the simplex algorithm, which moves from vertex to vertex. It computes the search direction as an orthogonal projection of the dual objective gradient onto a relevant null space. In each iteration, it solves a single small triangular system, compared with four triangular systems handled by the simplex algorithm. We report preliminary but favorable computational results with a set of standard Netlib test problems.

1. Introduction1.1. The State of the Art.• Simplex algorithm (G. B. Dantzig 1947) • Ellipsoid algorithm ( Khachiyan 1979)• Interior-point algorithm ( Karmarkar 1984) Pan’s work : • The obtuse-angle principle(1990)• Bisection simplex algorithm (1991)• The most-obtuse-angle rules(1997)• Deficient basis & projective pivot algorithms (1997-)• Nested pricing, Largest-distance pricing (2008-) • Affine-scaling pivot algorithm (submitted)

1.2. Basic Idea : Moving from face to face.

1.3. Model.

We are concerned with the linear

programming (LP) problem in the standard form:

The dual problem:

1.4. Model Reformulation. Program (1.2) can be transformed to:

2. Dual search direction and line search.

Dual face associated with B:

According Subprogram:

2.1. dual search direction

• The solution can be obtained by solving the upper triangular system alone:

2.3. Line search

2.4. Contracting dual face and updating the

Cholesky factor.

3.Optimality test.

3.1. Expanding dual face and updating the

Cholesky factor.

4. The dual face algorithm

An attractive feature: in each iteration, it solves

a single smaller triangular system.

4.1. More features.

Any expansion iteration must be followed by a contraction

iteration. The latter is called accompanying (contraction)

iteration.

If a component (excluding one that just be set to zero in a

preceding expansion iteration) of is zero, it is termed

degenerate.

5. Phase-1

Auxiliary program:

where

6. Computational results.

DFA: the proposed algorithm.

RSA: dense implementation of the standard revised

simplex algorithm.

Both primal and dual feasibility tolerance were taken to be , and row

relative tolerance was .

Compiled using the Visual FORTRAN 5.0, code DFA and RSA were run

under a Windows XP system Home Edition Version 2002 on an IBM PC

with an Intel(R) Pentium(R) processor 1.00GB of 1.86GHz memory,

and about 16 digits of precision. All reported CPU times were

measured in seconds with utility routine CPU_TIME.

Tested were a set of 26 standard problems from NETLIB.

Ratio of RSA to DFA