I . I . Dikin ' s Convergence Result for the Affine - Scaling Algorithm

The affine-scaling algorithm is an analogue of Karmarkar's linear programming algorithm that uses affine transformations instead of projective transformations. Although this variant lacks some of the nice properties of Karmarkar's algorithm (for example, it is probably not a polynomial-time algorithm). it nevertheless performs well in computer implementations. It has recently come to the attention of the western mathematical programming community that a Soviet mathematician, I. I. Dikin, proposed the basic affine~scaling algorithm in 1967 and published a proof of convergence in 1974. Dikin's convergence proof assumes only primal nondegeneracy, while all other known proofs require both primal and dual nondegeneracy. Our aim in this paper is to give a clear presentation of Dikin's ideas.