Rescaled coordinate descent methods for linear programming

We propose two simple polynomial-time algorithms to find a positive solution to Ax=0Ax=0 . Both algorithms iterate between coordinate descent steps similar to von Neumann’s algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax=0Ax=0 , x≥0x≥0 . This is an extended abstract. The missing proofs will be provided in the full version

Rescaling algorithms for linear conic feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ R m× n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of A T. Both algorithms iterate between simple first-order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure ρ A is negative, then the kernel problem is feasible, and the worst-case complexity of the kernel algorithm is O((m 3n + mn 2)log|ρ A| −1); if ρ A > 0, then the image problem is feasible, and the image algorithm runs in time O(m 2n 2 log ρ A −1). We also extend the image algorithm to the oracle setting. We address the degenerate case ρA = 0 by extending our algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A T. In this case, the running time bounds are expressed in the bit-size model of computation: for an input matrix A with integer entries and total encoding length L, the maximum support kernel algorithm runs in time O((m 3n + mn 2)L), whereas the maximum support image algorithm runs in time O(m 2n 2L). The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for linear programming

Rescaled coordinate descent methods for linear programming

We propose two simple polynomial-time algorithms to find a positive solution to Ax = 0. Both algorithms iterate between coordinate descent steps similar to von Neumann’s algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax =0, x ≥ 0. This is an extended abstract. The missing proofs will be provided in the full versio