Constraint Qualifications and Optimality Conditions for Multiobjective Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds

In this paper, we investigate constraint qualifications and optimality conditions for multiobjective mathematical programming problems with vanishing constraints (MOMPVC) on Hadamard manifolds. The MOMPVC-tailored generalized Guignard constraint qualification (MOMPVC-GGCQ) for MOMPVC is introduced in the setting of Hadamard manifolds. By employing MOMPVC-GGCQ and the intrinsic properties of Hadamard manifolds, we establish Karush–Kuhn–Tucker (KKT)-type necessary Pareto efficiency criteria for MOMPVC. Moreover, we introduce several MOMPVC-tailored constraint qualifications and develop interrelations among them. In particular, we establish that the MOMPVC-tailored constraint qualifications introduced in this paper turn out to be sufficient conditions for MOMPVC-GGCQ. Suitable illustrative examples are furnished in the framework of well-known Hadamard manifolds to validate and demonstrate the importance and significance of the derived results. To the best of our knowledge, this is the first time that constraint qualifications, their interrelations, and optimality criteria for MOMPVC have been explored in the framework of Hadamard manifolds.