An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming

In this paper, we extend Chubanov's new polynomial-time algorithm for linear programming to second-order cone programming based on the idea of cutting plane method. The algorithm finds an (Formula presented.) -dimensional vector x which satisfies Ax = 0, x ∈ K, where (Formula presented.) and K is a direct product of n second-order cones and half lines. Like Chubanov's algorithm, one iteration of the proposed algorithm consists of two phases: execution of a basic procedure and scaling. Within O(n log ∈⁻¹) calls of the basic procedure, the algorithm either (i) finds an interior feasible solution, (ii) finds a non-zero dual feasible solution, or (iii) verifies that there is no interior feasible solution whose minimum eigenvalue is greater than or equal to ϵ. Each basic procedure requires (Formula presented.) arithmetic operations, where d_i is the dimension of each second-order cone. If the problem is interior feasible, then the algorithm finds an interior feasible solution in O(n log cond(A,K)) calls of the basic procedure, where cond(A,K) is a condition number associated with the system.