Constraint Qualifications and Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Programming Problems with Vanishing Constraints on Hadamard Manifolds

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Abstract
This article investigates a class of nonsmooth interval-valued multiobjective programming problems with vanishing constraints (NIMPPVCs) on Hadamard manifolds, under locally Lipschitz continuity assumptions on objective and constraint functions. We establish that the various standard constraint qualifications, namely Cottle-type, Slater-type, Mangasarian-Fromovitz-type, and linearly independent constraint qualifications, usually violate at any feasible point of NIMPPVC. Moreover, we introduce several NIMPPVC-tailored constraint qualifications for NIMPPVC, in particular, Abadie constraint qualification (ACQ-VC), generalized Abadie constraint qualification (GACQ-VC), generalized Guignard constraint qualification (GGCQ-VC), Cottle-type constraint qualification (CCQ-VC), Slater-type constraint qualification (SCQ-VC), Mangasarian-Fromovitz-type constraint qualification (MFCQ-VC), and linearly independent constraint qualification (LICQ-VC), and further establish interrelations among them. In addition to this, by employing GGCQ-VC, we establish the Karush-Kuhn-Tucker (KKT)-type necessary optimality conditions for LR-efficient solutions of NIMPPVC via Clarke subdifferentials. Moreover, sufficient criteria of optimality for NIMPPVC are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Furthermore, the sufficient optimality conditions established in this paper are applied to propose an algorithm for identifying the LR-efficient solutions of NIMPPVC. Various illustrative examples on Hadamard manifolds are furnished to highlight the significance of the results derived in this paper. To the best of our knowledge, constraint qualifications and optimality conditions for NIMPPVC have been investigated for the first time in the Hadamard manifold framework.

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