This paper introduces a method for computing points satisfying the second-order necessary optimality conditions in constrained nonconvex minimization. The method comprises two independent steps corresponding to the first and second order conditions. The first-order step is a generic closed map algorithm which can be chosen from a variety of first-order algorithms, making it The second-order step can be viewed as a second-order minimization. We prove that any limit point of the resulting scheme satisfies the second-order necessary optimality condition, and establish the scheme's convergence rate and complexity, under standard and mild assumptions. Numerical tests validate our theoretical results, and illustrate how and when the proposed method can be efficiently implemented.