Rescaling Algorithms for Linear Conic Feasibility

TL;DR

This paper introduces polynomial-time rescaling algorithms for linear conic feasibility problems, addressing both feasible and degenerate cases, with applications to linear programming.

Contribution

It presents novel rescaling algorithms for kernel and image problems, extending to degenerate cases and providing polynomial-time solutions for linear programming feasibility.

Findings

Algorithms run in polynomial time for feasible cases.

Extended algorithms handle degenerate cases with maximum support solutions.

Applications include efficient linear programming feasibility testing.

Abstract

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \mathbb{R}^{m\times n}$, the kernel problem requires a positive vector in the kernel of $A$, and the image problem requires a positive vector in the image of $A^\top$. Both algorithms iterate between simple first order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure $\rho_A$ is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is $O\left((m^3n+mn^2)\log{|\rho_A|^{-1}}\right)$; if $\rho_A>0$, then the image problem is feasible and the image algorithm runs in time $O\left(m^2n^2\log{\rho_A^{-1}}\right)$. We also extend the image algorithm to the oracle setting. We address the degenerate case $\rho_A=0$ by extending our algorithms to find maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^\top$. 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 $O\left((m^3n+mn^2)L\right)$, while the maximum support image algorithm runs in time $O\left(m^2n^2L\right)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.