A note on a potential reduction algorithm for LP with simultaneous primal-dual updating

Abstract

Potential function reduction algorithms for linear programming and the linear complementarity problem use key projections p x and p s which are derived from the ‘double’ potential function, φ( x, s) = ø ln( x T s)− Σ j = 1 n ln( x j s j ), where x and s are primal and dual slacks vectors. For non-symmetric LP duality we show that the existence of s, y, x satisfying s = c − A T y, A x = b such that p x = (ϱ/x T s) X s − e and p s = (ϱ/x T s)S x − e yields simultaneous primal and dual projection-based updating during the process of reducing the potential function ø. The role of x, s in an O(√nL) simultaneous primal-dual update algorithm is discussed.