On Second-order Conditions for Quasiconvexity and Pseudoconvexity of $\mathcal{C}^{1,1}$-smooth Functions

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 Fr\'echet and Mordukhovich second-order subdifferentials.