In this paper, a nonconvex vector optimization problem with both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are E-differentiable. The so-called E-Fritz John necessary optimality conditions and the so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable multiobjective programming problems with both inequality and equality constraints. Further, the sufficient optimality conditions are derived for such nonconvex nonsmooth vector optimization problems under (generalized) E-convexity. The so-called vector E-Wolfe dual problem is defined for the considered E-differentiable multiobjective programming problem with both inequality and equality constraints and several dual theorems are established also under (generalized) E-convexity hypotheses.

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems. For an E-differentiable function, the concept of E-invexity is introduced as a generalization of the E-differentiable E-convexity notion. In addition, some properties of E-differentiable E-invex functions are investigated. Furthermore, the so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable vector optimization problems with both inequality and equality constraints. Also, the sufficiency of the E-Karush-Kuhn-Tucker necessary optimality conditions are proved for such E-differentiable vector optimization problems in which the involved functions are E-invex and/or generalized E-invex.

A new class of (not necessarily differentiable) multiobjective fractional programming problems with E-differentiable functions is considered. The so-called parametric E-Karush-Kuhn-Tucker necessary optimality conditions and, under E-convexity hypotheses, sufficient E-optimality conditions are established for such nonsmooth vector optimization problems. Further, various duality models are formulated for the considered E-differentiable multiobjective fractional programming problems and several E-duality results are derived also under appropriate E-convexity hypotheses.

<p style='text-indent:20px;'>In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems with <inline-formula><tex-math id="M4">\begin{document}$ E $\end{document}</tex-math></inline-formula>-differentiable functions. Namely, for an <inline-formula><tex-math id="M5">\begin{document}$ E $\end{document}</tex-math></inline-formula>-differentiable vector-valued function, the concept of <inline-formula><tex-math id="M6">\begin{document}$ V $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M7">\begin{document}$ E $\end{document}</tex-math></inline-formula>-invexity is defined as a generalization of the <inline-formula><tex-math id="M8">\begin{document}$ E $\end{document}</tex-math></inline-formula>-differentiable <inline-formula><tex-math id="M9">\begin{document}$ E $\end{document}</tex-math></inline-formula>-invexity notion and the concept of <inline-formula><tex-math id="M10">\begin{document}$ V $\end{document}</tex-math></inline-formula>-invexity. Further, the sufficiency of the so-called <inline-formula><tex-math id="M11">\begin{document}$ E $\end{document}</tex-math></inline-formula>-Karush-Kuhn-Tucker optimality conditions are established for the considered <inline-formula><tex-math id="M12">\begin{document}$ E $\end{document}</tex-math></inline-formula>-differentiable vector optimization problems with both inequality and equality constraints under <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M14">\begin{document}$ E $\end{document}</tex-math></inline-formula>-invexity hypotheses. Furthermore, the so-called vector <inline-formula><tex-math id="M15">\begin{document}$ E $\end{document}</tex-math></inline-formula>-dual problem in the sense of Mond-Weir is defined for the considered <inline-formula><tex-math id="M16">\begin{document}$ E $\end{document}</tex-math></inline-formula>-differentiable multiobjective programming problem and several <inline-formula><tex-math id="M17">\begin{document}$ E $\end{document}</tex-math></inline-formula>-duality theorems are derived also under appropriate <inline-formula><tex-math id="M18">\begin{document}$ V $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M19">\begin{document}$ E $\end{document}</tex-math></inline-formula>-invexity assumptions.</p>

In this paper, a nonconvex vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are E-differentiable. The so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable vector optimization problem with the multiple interval-valued objective function. Also the sufficient optimality conditions are derived for such intervalvalued vector optimization problems under appropriate (generalized) E-convexity hypotheses.