A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule

The dual Phase-1 algorithm using the most-obtuse-angle row pivot rule is very efficient for providing a dual feasible basis, in either the classical or the basis-deficiency-allowing context. In this paper, we establish a basis-deficiency-allowing Phase-I algorithm using the so-called most-obtuse-angle column pivot rule to produce a primal (deficient or full) basis. Our computational experiments with the smallest test problems from the standard NETLIB set show that a dense projected-gradient implementation largely outperforms that of the variation of the primal simplex method from the commercial code MATLAB LINPROG v1.17, and that a sparse projected-gradient implementation of a normalized revised version of the proposed algorithm runs 34% faster than the sparse implementation of the primal simplex method included in the commercial code TOMLAB LPSOLVE v3.0.