On Characterizing the Solution Sets of Pseudoinvex Extremum Problems

In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla f(x)^{T}\eta(\bar{x},x)=0$\end{document} , for each feasible point x, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar{x}$\end{document} is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995).