Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives

For a $$\mathcal {C}^2$$-smooth function on a finite-dimensional space, a necessary condition for its quasiconvexity is the positive semidefiniteness of its Hessian matrix on the subspace orthogonal to its gradient, whereas a sufficient condition for its strict pseudoconvexity is the positive definiteness of its Hessian matrix on the subspace orthogonal to its gradient. Our aim in this paper is to extend those conditions for $$\mathcal {C}^{1,1}$$-smooth functions by using the Frechet and Mordukhovich second-order subdifferentials.