A deficient-basis dual counterpart of Paparrizos, Samaras and Stephanides' primal-dual simplex-type algorithm

In a recent primal-dual simplex-type algorithm [K. Paparrizos, N. Samaras and G. Stephanides, 2003, A new efficient primal dual simplex algorithm. Computers & Operations Research 30, 1383–1399], their authors show how to take advantage of the knowledge of a primal feasible point and they work with a square basis during the whole process. In this paper we address what could be thought of as its deficient-basis dual counterpart by showing how to take advantage of the knowledge of a dual feasible point in a deficient-basis simplex-type environment. Three small examples are given to illustrate how the specific pivoting rules designed for the proposed algorithm deal with unbounded dual objectives, non-unique dual solutions and a classical exponential example by Goldfarb, thus avoiding some caveats of the dual simplex method. No computational results are reported, but we sketch some details of a sparse projected-gradient implementation in terms of Davis and Hager’s cholmod sparse Cholesky factorization (which is row updatable/downdatable) to solve the underlying least-squares subproblems, namely linear least-squares problems and projections onto linear manifolds.