## 1 Introduction
Robust optimization aims to find solutions that remain optimal even when faced with uncertainties in the problem's parameters. Generally, robust optimization reformulates the original problem into a deterministic equivalent, called the robust counterpart. This allows us to use the standard optimization algorithms to find solutions that are resilient to these uncertainties. Such problems arise frequently in various areas of modern research (see, for instance, [49; 50]). To tackle the barriers posed by the nonsmooth nature of data uncertainty problems, several authors have thoroughly investigated nonsmooth optimization problems with uncertain data under various assumptions. [PERSON] and [PERSON] [51] established nonsmooth optimality criteria for robust MOP. [PERSON] et al. [52] developed sufficient optimality criteria and dual models for nonsmooth MOP with uncertain data. [PERSON] et al. [53] introduced two new classes of generalized convex functions, which are not necessarily convex, and established optimality and duality results under the assumptions of generalized convexity for nonsmooth MOP with uncertain data. Recently, [PERSON] and [PERSON] [54] established necessary optimality conditions for robust nonsmooth MOP. Utilizing convexificators, [PERSON] et al. [55] deduced weak Pareto efficiency criteria and duality results for nonconvex MOP with uncertain data. It is worthwhile to mention that the optimality conditions and duality theorems have so far been studied by several researchers for nonsmooth robust optimization problems defined in Euclidean spaces (see, for instance, [52; 53; 54]). However, to the best of our knowledge, prior work has not addressed problems involving vanishing constraints and data uncertainty. In this paper, our main focus is to primarily tackle the aforementioned research gap. Motivated by the research work presented in [40; 45; 53; 54], we examine a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty via convexificators. We introduce GS-ACQ for UNMPVC and introduce generalized robust versions of various stationary points, namely RW-stationary,RM-stationary, and RS-stationary points, for UNMPVC. Subsequently, under generalized convexity assumptions, we deduce sufficient optimality conditions for UNMPVC. We further enrich the analysis of the primal problem, UNMPVC, by formulating WRD and MWRD-type dual models. This enables us to derive weak, strong, and strict converse duality relationships between the original problem and its dual model. The importance of results established in this paper is demonstrated through suitable illustrative examples. The fundamental contributions and novelty of this paper are twofold. At first, we introduce GS-ACQ and robust stationary conditions for UNMPVC, which is employed to extend the optimality conditions derived in [40] for nonsmooth MPVC to a more general category of optimization problems dealing with data uncertainty. Moreover, we extend the duality theorems established in [45] for MPVC to a wider class of mathematical programming problems dealing with data uncertainty. Secondly, we generalize the robust optimality and duality results established in [54; 55] to a broader class of general robust mathematical programming problems incorporating vanishing constraints. To the best of our knowledge, optimality and duality results in terms of convexificators presented in this paper have not been given before. The present article is structured as follows: we review some fundamental definitions and mathematical concepts in Section 2. In Section 3, we introduce GS-ACQ constraint qualification and various stationary conditions for UNMPVC. Moreover, we deduce sufficient optimality conditions for UNMPVC. In Sections 4 and 5, we formulate two dual models, namely WRD and MWRD, related to the problem of UNMPVC. We deduce various duality relationships between the UNMPVC and its corresponding dual models. In Section 6, we discuss the applicability of the results derived in this article in truss topology optimization.