Rescaling Algorithms for Linear Programming - Part I: Conic feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ Rm×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. 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 Goffin’s condition measure ρ̂A is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is O ( (m3n+mn2) log |ρ̂A|−1 ) ; if ρ̂A > 0, then the image problem is feasible and the image algorithm runs in time O ( m2n2 log ρ̂−1 A ) . We also address the degenerate case ρ̂A = 0: we extend our algorithms for finding maximum support nonnegative vectors in the kernel of A and in the image of A⊤. We obtain the same running time bounds, with ρ̂A 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 O ( (m3n+mn2)L ) , whereas both image algorithms run in timeO ( m2n2L ) . The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.