In this paper, we consider a multiobjective programming problem with inequality and set constraints. We derive sufficient conditions for the optimality of a feasible point under generalized invexity assumptions in terms of convexificators. We give an example to illustrate that the concept of invexity in terms of convexificators is weaker than invexity in terms of other subdifferentials. We formulate Wolfe and Mond-Weir type duals for the nonsmooth multiobjective programming problem with inequality and set constraints in terms of convexificators. We establish weak, strong, converse, restricted converse and strict converse duality results under the assumptions of invexity and strict invexity using convexificators between the primal and the Wolfe dual. We derive the respective results between the primal and the Mond-Weir dual under the assumptions of generalized pseudoinvexity, strict pseudoinvexity and quasiinvexity in terms of convexificators. We also derive the relationship between a weak vector saddle-point and a weakly efficient solution of the multiobjective programming problem.