Rescaling Algorithms for Linear Programming - Part I: Conic feasibility.

propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $Ain mathbb{R}^{mtimes n}$, 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 steps improve natural geometric potentials in the domain and image spaces, respectively. If Goffinu0027s condition measure $hat rho_A$ is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is $Oleft((m^3n+mn^2)log{|hat rho_A|^{-1}}right)$; if $hatrho_Au003e0$, then the image problem is feasible and the image algorithm runs in time $Oleft(m^2n^2log{hat rho_A^{-1}}right)$. We also address the degenerate case $hatrho_A=0$: we extend our algorithms for finding maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^top$. obtain the same running time bounds, with $hatrho_A$ replaced by appropriate condition numbers. In case the input matrix $A$ has integer entries and total encoding length $L$, all algorithms are polynomial. Both full support and maximum support kernel algorithms run in time $Oleft((m^3n+mn^2)Lright)$, whereas both image algorithms run in time $Oleft(m^2n^2Lright)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.