On the Weak Second-order Optimality Condition for Nonlinear Semidefinite and Second-order Cone Programming

Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish condition of such type two classes problems, namely semidefinite second-order cone programming, assuming Robinson’s constraint qualification weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via penalty-based strategy, aimed at providing strong global convergence results first- algorithms. Since not complementarity, the critical does reduce to subspace, thus, arrive defined in terms lineality space cone. case reduces standard widely used as stationarity measure algorithmic practice.