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

Abstract

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 1-A. 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.

On the chaotic behavior of the primal–dual affine–scaling algorithm for linear optimization

“…If θ > 2/3 then h has a unique fixed point r ∈ (0, 1), which can be found by solving the cubic equation h(x) = x that we already encountered in equation (9). This cubic is divisible by x − 1, so we find that r satisfies the quadratic equation…”

A simple proof of a primal affine scaling method

“…Stimulated by [6], but independently, Tsuchiya and Muramatsu [16] proved that the method converges to the optimum solution if the step size is restricted to 2/3, and that in this case, the primal sequence converges to the relative interior of the optimal face of the primal and the dual sequence to the analytic center of the optimal face of the dual. Hall and Vanderbei [8] present an example that shows the 2/3 is tight in the sense that with a larger step size, the dual sequence may not converge.…”

“…In the series of papers (Tsuchiya [14,15], Dikin [5,6] and Tsuchiya and Muramatsu [16]), convergence is shown when the step size is restricted to two-thirds of the distance to the boundary. Prior to these results, convergence of the primal sequence was known, but its optimality was only known under non-degeneracy, see for example Vanderbei and Lagarias [18].…”

Superlinear convergence of the affine scaling algorithm

“…Recently, Dikin [8] and Tsuchiya and Muramatsu [29] proved global convergence of the long-step version of the AS algorithm [31] for degenerate LP problems. This longstep version is the one in which the next iterate is determined by taking a fixed fi'action A ~ (0, I) of the whole step to the boundary of the feasible region.…”

“…Because of its theoretical and practical importance, there are a number of papers which study its global and local convergence [4,[6][7][8]12,16,25,26,[28][29][30][31] as well as its continuous trajectories [3, 16,32]. For computational experiments and implementation issues related to the AS algorithm, we refer the reader to [ 1,2,5, 18,21,22].…”