Quadratic convergence of a smoothing Newton method for symmetric cone programming without strict complementarity

Abstract

This paper deals with symmetric cone programming (SCP), which includes the linear programming (LP), the second-order cone programming (SOCP), the semidefinite programming (SDP) as special cases. Based on the Chen–Mangasarian smoothing function of the projection operator onto symmetric cones, we establish a smoothing Newton method for SCP. Global and quadratic convergence of the proposed algorithm is established under the primal and dual constraint nondegeneracies, but without the strict complementarity. Moreover, we show the equivalence at a Karush–Kuhn–Tucker (KKT) point among the primal and dual constraint nondegeneracies, the nonsingularity of the B-subdifferential of the smoothing counterpart of the KKT system, and the nonsingularity of the corresponding Clarke’s generalized Jacobian.