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 3 n + m n 2 ) l o g | ρ A | − 1 ) ; if ρ A > 0 , then the image problem is feasible, and the image algorithm runs in time O ( m 2 n 2 ⁡ l o g ⁡ ρ A − 1 ) . We also extend the image algorithm to the oracle setting. We address the degenerate case ρ A = 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 3 n + m n 2 ) L ) , whereas the maximum support image algorithm runs in time O ( m 2 n 2 L ) . The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for linear programming.