Accelerated Affine Scaling Algorithms for Linear Programming Problems.

Interior point methods are widely used to solve linear programming problems. In this work, we present two accelerated primal affine scaling algorithms to achieve faster convergence for solving linear programming problems. For the first algorithm, we integrate nesterov's acceleration in the primal affine scaling method with an acceleration parameter $0 \leq \beta <1$, which in turn generalizes the original primal affine scaling method (no acceleration, $\beta =0$). We provide proof of convergence of the proposed algorithm considering long step size. We also prove the convergence of priaml-dual sequence without the degeneracy assumption given that $\alpha, \beta \in Q, \ Q = \{(\alpha, \beta) | \ 0 < \alpha < 1 ,\ 0 \leq \beta < \frac{1}{\phi}, \ \alpha +\beta \leq \frac{2}{3} \}$, where $\phi = 1.618...$ is the so called golden ratio. Then we introduce a second algorithm to further accelerate the convergence rate of the first variant by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal affine scaling method.