Interval-valued value function and its application in interval optimization problems

Abstract

In this article, we attempt to propose the concept of interval-valued value function. To propose the concept, we study the notion of a Lagrangian interval-valued function (IVF). Subsequently, a weak duality theorem for interval optimization problem (IOP) is derived, by which a relation between the saddle point and efficient solutions of the corresponding IOP is found. Furthermore, a characterization of the gH-subdifferential set of an interval-valued value function is given. A saddle point efficiency interpretation of an interval-valued value function is studied, which is based on the saddle point criterion of the Lagrangian IVF. In the sequel, a relation of gH-subdifferential set of an interval-valued value function with the stability of a solution to an IOP is shown. Furthermore, the gH-subdifferential set of an interval-valued value function is used to prove a relation concerning the efficiency of a solution to an IOP under certain restrictions. Finally, an example to show an application of interval-valued value function in a practical phenomenon is discussed. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.