The authors study the relations between vector variational inequalities and nonsmooth multi-objective optimization problems in the sense of strict minimizers of higher order. They introduce an extension of the concept of higher-order strong pseudoconvexity for Lipschitz functions, which they call {\it higher-order strong pseudoconvexity of type I}. Also they provide some examples that illustrate the applicability of their results. Finally, they identify (under suitable hypotheses) strict higher-order minimizers, vector critical points and solutions of the weak vector variational inequality problem under higher-order strong pseudoconvexity of type I.