Existence Results and KKT Optimality Conditions for Generalized Quasiconvex Functions

Abstract: We studied a new notion of generalized convex functions called $e$-quasi-con-ve-xi-ty, which encompasses both quasiconvex and $e$-convex functions, including all Lipschitz functions. By extending the standard properties of quasiconvex functions to $e$-quasiconvex functions, we establish sufficient conditions for the nonemptiness and compactness of the solution set when minimizing an $e$-quasiconvex function, leveraging generalized asymptotic functions, a result which remains applicable even when the set of minimizers is nonconvex. Furthermore, in the differentiable case, we ensure the sufficiency of the KKT optimality conditions when the constraint functions in the mathematical programming problems are $e$-quasiconvex. Finally, we illustrate our new results with several nonconvex (non-quasiconvex) examples.