Primal-Dual Affine Scaling Methods for Linear Problems

Abstract

The introduction of Karmarkar’s polynomial time algorithm for linear programming (LP) [126] in 1984 is responsible for an enormous production of research papers on LP in the last decade. Two of the early papers were Barnes [16] and Vanderbei et al. [238], who independently proposed the affine scaling algorithm as a simplification of Karmarkar’s algorithm. While Karmarkar performed a projective scaling in each iteration, this algorithm used a simpler affine scaling. Convergence of the iterates to optimality was shown (under nondegeneracy assumptions), but no complexity proof was given. Implementations soon showed that the affine scaling algorithm was very efficient, see Adler et al. [1]. After the publication of Vanderbei’s paper, the editor of Mathematical Programming received a letter [44] from the Russian researcher Dikin, revealing that the affine scaling algorithm was already proposed by him in 1967 [42], see also [43]. Around the same time a paper by Gill et al. [71] showed that Karmarkar’s algorithm has close connections with the logarithmic barrier method, introduced by Frisch [62] in 1955 and extensively investigated by Fiacco and McCormick [53] among others.