## 1 Introduction
[PERSON] and [PERSON] [25] employed convexificators to establish constraint qualifications (CQ) for nonsmooth optimization problems with mixed constraints. Over the last few years, mathematical programming problems with vanishing constraints (MPVC) have gained prominence as a significant area of research (see, [26; 27; 28]). [PERSON] and [PERSON] [29] are considered the pioneers of MPVC, laying the groundwork for its formulation and development. The nomenclature \"vanishing constraints\" aptly captures a critical feature of these problems. In many real-world applications, certain constraints become inactive for specific feasible solutions within the decision space (see, [30]). A significant challenge in investigating MPVC is the fact that the feasible set may be non-convex and disconnected, even when all constraint functions are convex (see [30]). [PERSON] and [PERSON] [31] made significant contributions to MPVC by deriving necessary optimality criteria for MPVC. [PERSON] and [PERSON] [32] introduced various MPVC-tailored CQs and established interrelations among them. Later, [PERSON] and [PERSON] [33] investigated Guignard and Abadie CQs for MPVC. [PERSON] and [PERSON] [34] introduced various CQs for MPVC with non-differentiable constraints and established several KKT-type stationarity conditions. For nonsmooth MPVC, [PERSON] et al. [35] estimated the Frechet normal cone of the feasible set and derived corresponding stationarity conditions. [PERSON] et al. [36] introduced two new Abadie-type CQ and established the necessary optimality criteria for properly efficient solutions for multiobjective MPVC with non-differentiable convex constraints and continuously differentiable objective functions. [PERSON] et al. [37] established first-order necessary and sufficient optimality criteria for an efficient and weakly efficient solution for nonsmooth multiobjective MPVC. [PERSON] and [PERSON] [38] introduced several CQs for MPVC on Hadamard manifolds. Later, [PERSON] et al. [39] introduced several CQs and established Pareto efficiency criteria for multiobjective MPVC based on Hadamard manifolds. Using convexificators, [PERSON] et al. [40] introduced CQs and stationary conditions for MPVC involving non-differentiable functions. By leveraging the concept of convexificators, [PERSON] et al. [41] introduced generalized cottle-type and Guignard-type CQs for nonsmooth semidefinite multiobjective MPVC and derived stationary conditions for the considered problem. Recently, [PERSON] et al. [42] introduced the concept of convexificators within the context of Hadamard manifolds and employed them to investigate multiobjective MPVC on Hadamard manifolds. Duality is a fundamental concept in optimization that allows us to view the same optimization problem from a different perspective. In many circumstances, tackling the dual problem is often seen to be more advantageous than addressing the primal problem. [PERSON] [43] and [PERSON] and [PERSON] [44] introduced two important dual models for differentiable scalar functions, which have subsequently been extended to nonsmooth scalar and multiobjective optimization problems under various generalized convexity and subdifferential assumptions. [PERSON] et al. [45] presented Mond-Weir- and Wolfe-type dual models for MPVC. [PERSON] et al. [46] formulated new Mond-Weir- and Wolfe-type dual models for MPVC, which eliminate the need for index set calculations. [PERSON] [47] studied Pareto efficiency criteria and various duality results for multiobjective semi-infinite programming problems with vanishing constraints. Recently, in the setting of Hadamard manifolds, for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints, [PERSON] et al. [48] investigated optimality criteria and formulated dual models. Robust optimization deals with uncertain data in an optimization problem. Mathematical programming problems often incorporate parameters with inherent uncertainty due to estimation errors, rounding discrepancies, or implementation issues.