Abstract
Degeneracy can cause enormous problems when solving large scale linear programming problems. This is not only because there is a possibility that the problem can cycle, but also because a large number of iterations can be executed that do not improve the objective. In this article a procedure which utilises derived reduced costs is discussed. The derived reduced cost of a non- basic variable is defined in such a way that it makes the introduction to the non-basic variable into the basis unattractive if such a decision fails to improve the objective. The procedure deliberately strives to combat degeneracy using derived reduced costs, but it also utilises the advantageous properties of the classical gradient methods.
Highlights
The solution of any programming problem consists of the following two choices for moving from one extreme point to another: the direction which is taken, and the length of the step
In linear programming problems the direction of movement is determined by the choice of the pivot column, while the length of the step is influenced by both the choice of the pivot column and pivot row
Wolfe and Cutler [20] have suggested that should the selection of a pivot row be at issue, the pivot row should be selected such that the pivot is as large as possible
Summary
The solution of any programming problem consists of the following two choices for moving from one extreme point to another: the direction which is taken, and the length of the step. In linear programming problems the direction of movement is determined by the choice of the pivot column, while the length of the step is influenced by both the choice of the pivot column and pivot row. Wolfe and Cutler [20] have suggested that should the selection of a pivot row be at issue, the pivot row should be selected such that the pivot is as large as possible. A method [15] was developed in which the influence of the gradient methods and the derived reduced costs can be controlled.
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