Normal and tangent cones for set of intervals and their application in optimization with functions of interval variables

Abstract

In this article, we attempt to characterize optimum solutions for optimization problems with interval-valued functions of interval variables. As the constraint set or the underlying variable spaces of such optimization problems are a set of intervals, we introduce, analyze, and interrelate the notions of normal cone and tangent cone of a set of intervals. In the sequel, their various kinds of properties are defined, such as closedness, weak intersection rule, some algebraic preserving properties, etc. The dual correspondence between the tangent and normal cones is also analyzed. For constrained optimization problems, the normal cone is used to characterize efficient solutions. Furthermore, we derive a necessary condition for efficient solutions for interval optimization problems with the help of Lagrange multipliers, tangent cones, and normal cones. Lastly, an application of the proposed normal cone in solving an interval-valued support vector machines type problem is discussed. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.