On invex functions with the same mapping η in single-and multivalued nonsmooth optimization

In this paper, we investigate nonsmooth unconstrained single- and multiobjective optimization problems, in which the objective functions are locally Lipschitz continuous (LLC) and possibly nondifferentiable. Using the Clarke subdifferential, we study invexity properties of such functions and focus on the characteriza-tion of finite families of functions that are invex with respect to the same mapping η. We establish the necessary and sufficient optimality conditions for weak Pareto and Pareto optimal solutions under the assumption of V-invexity. Furthermore, we derive new equivalence results, linking the invexity of individual component functions with the invexity of their scalarized counterparts in the nonsmooth setting. We show that a straightforward extension of known differentiable results is not always possible and illustrate this limitation through a carefully constructed counterexample. The results obtained generalize the existing characterizations of invexity with respect to the same mapping to the class of nonsmooth LLC functions, thereby extending the applicability of invexity theory in nonsmooth and multiobjective optimization.