Nonsingularity in second-order cone programming via the smoothing metric projector

Abstract

Based on the differential properties of the smoothing metric projector onto the second-order cone, we prove that, for a locally optimal solution to a nonlinear second-order cone programming problem, the nonsingularity of the Clarke’s generalized Jacobian of the smoothing Karush-Kuhn-Tucker system, constructed by the smoothing metric projector, is equivalent to the strong second-order sufficient condition and constraint nondegeneracy, which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point. Moreover, this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point. Interestingly, the analysis does not need the strict complementarity condition.