In this paper, we extend the concept of quasidifferential to a new notion called semi-quasidifferential. This generalization is motivated by the convexificator notion. Some important properties of semi-quasidifferentials are established. The relationship between semi-quasidifferentials and the Clarke subdifferential is studied, and a mean value theorem in terms of semi-quasidifferentials is proved. It is shown that this notion is helpful to investigate nonsmooth optimization problems even when the objective and/or constraint functions are discontinuous. Considering a multiobjective optimization problem, a characterization of some cones related to the feasible set is provided. They are used for deriving necessary and sufficient optimality conditions. We close the paper by obtaining optimality conditions in multiobjective optimization in terms of semi-quasidifferentials. Some outcomes of the current work generalize the related results existing in the literature.