A Piecewise Linear Dual Phase-1 Algorithm for the Simplex Method

A dual phase-1 algorithm for the simplex method that handles all types of variables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The new method can be viewed as a generalization of traditional phase-1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. In addition to this main achievement it has some further important and favorable features, namely, it is very efficient in coping with degeneracy and numerical difficulties. Both theoretical and computational issues are addressed. Examples are also given that demonstrate the power and flexibility of the method.