Quasiconjugate duality and optimality conditions for quasiconvex optimization

In nonlinear optimization, conjugate functions and subdifferentials play an essential role. In particular, Fenchel conjugate is the most well known conjugate function in convex optimization. In quasiconvex optimization, extra parameters for quasiconjugate functions have been introduced in order to show duality theorems, for example lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-quasiconjugate and lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-semiconjugate. By these extra parameters, we can show duality results that hold for general quasiconvex objective functions. On the other hand, extra parameters usually increase the complexity of dual problems. Hence, conjugate functions without extra parameters have also been investigated, for example H-quasiconjugate, R-quasiconjugate, and so on. However, there are some open problems. In this paper, we study quasiconjugate duality and optimality conditions for quasiconvex optimization without extra parameters. We investigate three types of quasiconjugate dual problems, and show sufficient conditions for strong duality. We introduce three types of quasi-subdifferentials, and study optimality conditions and characterizations of the solution set. Additionally, we give a classification of quasiconvex optimization problems in terms of quasiconjugate duality.