O( nρL)-iteration and O( n3L)-operation potential reduction algorithms for linear programming

Abstract

In this paper we propose two potential reduction algorithms, which we call Algorithm 1 and Algorithm 2, for linear programming. Algorithm 1 has parameters θ in the potential function and β which determines a step size. Suppose that θ= n 1− σ and β=0.2 n 0.5- ϱ for 0.5⩽ θ⩽ ϱ⩽1. Then Algorithm 1 requires at most O( n ϱ L) iterations and O( n 3 L) arithmetic operations in total. If we take θ= n and β=0.2, Algorithm 1 is similar to Ye's primal form algorithm. Algorithm 2 has a parameter θ= n 1− σ for σ∈[0.5, 1]. It requires at most O( n σ L) iterations and O( n 3 L) arithmetic operations in total. If θ= n , Algorithm 2 is similar to the long step algorithm of Anstreicher and Bosch.