CHAPTER 2 A Primal-Dual Interior Point Algorithm for Linear Programming Masakazu Kojima, Shinji Mizuno, and Akiko Yoshise Abstract. This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 - Yf/n); each iteration reduces the duality gap by at least Yf/n. Here n denotes the size of the problems and Yf a positive number depending on initial interior feasible solutions of the problems. The algorithm is based on an application of the classical logarithmic barrier function method to primal and dual linear programs, which has recently been proposed and studied by Megiddo. §1. Introduction This chapter deals with the pair of the standard form linear program and its dual. (P) (D) Minimize c T x subject to XES = {x E R n : Ax = b, x O}. Maximize b T Y subject to (y,z) E T = {(y,z) E Rm+n: ATy + Z = c, Z Here R n denotes the n-dimensional Euclidean space, cERn and bERm are constant column vectors, and A is an m x n constant matrix. Throughout the chapter we impose the following assumptions on these problems: (a) The set Sf = {x E S : x> O} of the strictly positive feasible solutions of the primal problem (P) is nonempty. N. Megiddo (ed.), Progress in Mathematical Programming © Springer-Verlag New York Inc. 1989