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For a $\mathcal{C}^2$-smooth function on a finite-dimensional space, anecessary condition for its quasiconvexity is the positive semidefiniteness ofits Hessian matrix on the subspace orthogonal to its gradient, whereas asufficient condition for its strict pseudoconvexity is the positivedefiniteness 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 Mordukhovichsecond-order subdifferentials.

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