Steepest Edge , Degeneracy and Conditioning in LPbyR

Steepest edge criteria have often been shown to work well in Simplex or Active Set methods for Linear Programming (LP). It is therefore important that any technique for resolving degeneracy in LP should conveniently be able to use steepest edge criteria in the selection of the search direction. A recursive technique for resolving degeneracy 5] is of some interest in that it provides a guarantee of termination, even in the presence of round-oo errors. However it works in both primal and dual spaces and would need steepest edge coeecients in both spaces to take full advantage of steepest edge criteria. A new method of resolving degeneracy is described which provides a similar guarantee of termination and is readily implemented. The method works only in the primal space so that only one set of steepest edge coeecients needs to be maintained. It is also shown that the steepest edge coeecients provide information from which the expected conditioning of the current LP basis can be calculated cheaply. This provides a more accurate indication of the actual conditioning of a system than is obtained from norm-based condition numbers. A table of condition numbers for the SOL test set is presented and has some interesting implications for the solvability of LP problems when round-oo error is present. This idea also enables the conditioning of null space matrices to be estimated.