Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds

This article is concerned with nonsmooth multiobjective semi-infinite programming problems with vanishing constraints in the setting of Hadamard manifolds (abbreviated as, (NMSIPVC)). We present the Abadie constraint qualification (abbreviated as, (ACQ)) and a modified version of (ACQ), namely, (NMSIPVC)-tailored Abadie constraint qualification (abbreviated as, (NMSIPVC-ACQ)), for (NMSIPVC). The Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary criteria of optimality for (NMSIPVC) are derived by employing (ACQ) and (NMSIPVC-ACQ). Moreover, sufficient criteria of optimality for (NMSIPVC) are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Further, we formulate Wolfe type and Mond-Weir type dual models related to (NMSIPVC) and derive several duality results that relate (NMSIPVC) and the corresponding dual models. Non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. To the best of our knowledge, this is for the first time that constraint qualifications, optimality conditions, and duality results have been investigated for (NMSIPVC) in the setting of Hadamard manifolds.