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Let 𝔞 be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd_𝔞(M, N) denote the supremum of the i's such that . First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd_𝔞(M, N). Next, over a Cohen–Macaulay local ring (R, 𝔪), we show that provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne–Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.