Critical stationary points and descent from saddlepoints in constrained optimization via implicit automatic differentiation

A stationary point of a constrained optimization problem is called critical, if the first order necessary optimality conditions are fulfilled but the stationary point cannot be classified using second order optimality conditions. In this paper the problem of classifying a critical stationary point is reduced to the application of higher order optimality conditions for a special unconstrained optimization problem. This is possible using implicit automatic differentiation. Theoretical and computational aspects of descending from saddlepoints are considered.