A Study of Constrained Fractional Optimization Problems Involving Interval-Valued Functions

Abstract

This study explores the optimality criteria for a class of fractional optimization problems characterized by interval-valued objective functions under constraints. These problems involve curvilinear fractional integral cost functionals that are path-independent and fundamentally connected to fractional calculus, particularly through Riemann–Liouville fractional integrals. The application of fractional calculus is particularly significant in modeling complex systems exhibiting memory effects, as the fractional derivatives and integrals naturally incorporate the system’s historical behavior. An important outcome of this research is the development and demonstration of a core optimality condition—rooted in fractional calculus—ensuring that a solution locally LR-optimal for the associated variational control problem governed by PDE and PDI constraints is also globally LR-optimal. To underscore the practical relevance of these findings, an illustrative example is provided, demonstrating how fractional calculus and memory dynamics within complex systems can be effectively utilized to model and optimize real-world dynamic processes, specifically in the control of artificial neural networks.