This paper presents a variant of the primal affine scaling method, which it calls the primal power affine scaling method. This method is defined by choosing a real r>0.5, and is similar to the power barrier variant of the primal-dual homotopy methods considered by den Hertog, Roos and Terlaky and Sheu and Fang. Here, the paper analyzes the methods for r>1. The analysis for 0.50<r<1 is similar, and can be readily carried out with minor modifications. Under the non-degeneracy assumption, the paper shows that the method converges for any choice of the step size α. To analyze the convergence without the non-degeneracy assumption, it defines a power center of a polytope. The paper uses the connection of the computation of the power center by Newton’s method and the steps of the method to generalize the 2/3rd result of Tsuchiya and Muramatsu. It shows that with a constant step size α such that α/(1-α)2r<2/(2r-1) and with a variable asymptotic step size αk uniformly bounded away from 2/(2r+1), the primal sequence converges to the relative interior of the optimal primal face, and the dual sequence converges to the power center of the optimal dual face. The paper also presents an accelerated version of the method. It shows that the two-step superlinear convergence rate of the method is 1+r/(r+1), while the three-step convergence rate is 1+3r/(r+2). Using the measure of Ostrowski, the paper notes that the three-step method for r=4 is more efficient than the two-step quadratically convergent method, which is the limit of the two-step method as r approaches infinity.
