Alternative Approaches to Feasibility in Projective Methods for Linear Programming

Abstract

This paper concerns further developments in primal projective methods based on the potential function of De Ghellinck and Vial. Their method is unique in that feasibility and optimality are approached simultaneously from an interior starting point, without the addition of an artificial variable to handle feasibility, and without penalty or barrier terms. This behavior is obtained by incorporating the infeasibilities within the potential function. The primal algorithm requires an initial optimal objective estimate, and results are given demonstrating sensitivity of its value. We show that, by choosing a different initial search direction, feasibility and optimality can be treated in separate phases. Results are given, indicating that the two-phase method is often superior to the original single-phase method. We also describe a variant that selects the search direction on the basis of potential reduction, and a technique that allows the use of estimates that are not necessarily lower bounds for the optimal objective value. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.