A differentiable path-following algorithm for computing perfect stationary points

This paper is concerned with the computation of perfect stationary point, which is a strict refinement of stationary point. A differentiable homotopy method is developed for finding perfect stationary points of continuous functions on convex polytopes. We constitute an artificial problem by introducing a continuously differentiable function of an extra variable. With the optimality conditions of this problem and a fixed point argument, a differentiable homotopy mapping is constructed. As the extra variable becomes close to zero, the homotopy path naturally provides a sequence of closely approximate stationary points on perturbed polytopes, and converges to a perfect stationary point on the original polytope. Numerical experiments are implemented to further illustrate the effectiveness of our method.