The main research goal of this paper is to investigate the generalized convexity of interval-valued functions under the total order relation and apply it to a class of unconstrained interval-valued optimization problems. The paper is well written and structured, theorems are carefully proved, and the text is accompanied with simple and easy-to-understand examples.
After providing preliminaries, the authors first present a new definition of a preinvex interval-valued function under \(cw\)-order and prove that it is equivalent to a convex interval-valued function on \([0, 1]\). Then, a necessary and sufficient condition for it is given under the condition that the endpoint functions are differentiable. In the next section, a new concept of \(\preceq_{cw}\)-semicontinuity is introduced and its close relationship with preinvex interval-valued functions is proved. Finally, as an application related to preinvex interval-valued functions, a class of unconstrained interval-valued optimization problems is considered and existence theorems for optimal solutions through variational inequalities are derived.