Global Convergence of a Long-Step Affine Scaling Algorithm for Degenerate Linear Programming Problems

In this paper we present new global convergence results on a long-step affine scaling algorithm obtained by means of the local Karmarkar potential functions. This development was triggered by Dikin’s interesting result on the convergence of the dual estimates associated with a longstep affine scaling algorithm for homogeneous LP problems with unique optimal solutions. Without requiring any assumption on degeneracy, we show that moving a fixed proportion A up to two-thirds of the way to the boundary at each iteration ensures convergence of the iterates to an interior point of the optimal face as well as the dual estimates to the analytic center of the dual optimal face, where the asymptotic reduction rate of the value of the objective function is 1A. We also give an example showing that this result is tight to obtain convergence of the dual estimates to the analytic center of the dual optimal face. Key words, linear programming, interior point methods, affine scaling algorithm, global analysis, degenerate problems AMS subject classification. 90C05 0. Introduction. Since Karmarkar [17] proposed the projective scaling algorithm for linear programming in 1984, a number of interior point algorithms have been proposed and implemented. The affine scaling algorithm, originated by Dikin [7] and rediscovered by several authors including Barnes [4], Vanderbei, Meketon, and Freedman [43], Karmarkar and aamakrishnan [18], and Adler et al. [1], is one of the most popular interior point algorithms obtained by substituting the affine scaling transformation in place of the projective transformation in Karmarkar’s algorithm. This simple algorithm works well in practice, and now several promising experimental results [1], [2], [6], [12], [21], [24], [26], [30], [33] are reported. In contrast to its great performance in practice, our knowledge on this algorithm is rather poor, particularly under the existence of degeneracy, and there are large gaps between the theoretical convergence results and the existing efficient implementations. The first problem to be addressed is global convergence, which is one of the most fundamental properties to be shown from the theoretical point of view. There have been several milestone papers on this problem [3], [4], [8], [11], [14], [23], [34], [43], [44] under various step-size choices and nondegeneracy assumptions; see [15] for a survey. The analysis becomes difficult, particularly when we remove the primal nondegeneracy condition. We introduced the local Karmarkar potential function [39] to overcome the difficulty and then succeeded in proving the global convergence without any assumption on degeneracy with the step-size 1/8 [38], where the Dikin’s displacement vector is taken as the unit. This bound has been the best obtained so far theoretically, but it *Received by the editors FebrUary 24, 1992; accepted for publication (in revised form) January 19, 1994. This research was supported in part by a Grant-in-Aid for Encouragement of Young Scientists 03740126 (1991) and Overseas Research Scholars (1992) of the Ministry of Education, Science and Culture of Japan.. TThe Institute of Statistical Mathematics, 4-6-7 Minami-Azabu Minato-ku Tokyo 106 Japan (tsuchiya@sun312, ism. ac. jp). :Department of Mechanical Engineering, Sophia University, 7-1, Kioicho, Chiyoda-ku, Tokyo 102 Japan.