Necessary Optimality Conditions for Vector Reverse Convex Minimization Problems via a Conjugate Duality

In this paper, we are concerned with a vector reverse convex minimization problem (P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal {P})$\end{document}. For such a problem, by means of the so-called Fenchel–Lagrange duality, we provide necessary optimality conditions for proper efficiency in the sense of Geoffrion. This duality is used after a decomposition of problem (P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal {P})$\end{document} into a family of convex vector minimization subproblems and scalarization of these subproblems. The optimality conditions are expressed in terms of subdifferentials and normal cones in the sense of convex analysis. The obtained results are new in the literature of vector reverse convex programming. Moreover, some of them extend with improvement some similar results given in the literature, from the scalar case to the vectorial one.