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The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.