An exterior point method for computing points that satisfy second-order necessary conditions for a C^1,1 optimization problem

We obtain a second-order generalized chain rule for composite C1,1 functions using a generalized Hessian matrix. We introduce a unified exterior point penalty method for a C1,1 constrained minimization problem and derive second-order necessary conditions for exterior point penalty problem using the established generalized chain rule. We then show that any limiting point of the sequences obtained by the exterior point method satisfies the second-order necessary conditions.