Generalized Convexity via Clarke Subdifferential of Interval Mappings and Its Optimization Applications

Clarke subdifferential and directional derivative theories are well developed for real functions. However, the nonsmooth analysis framework for interval mappings is still incomplete. Few works reveal the inherent relationship between Clarke directional derivatives of interval mappings and their endpoint functions. This paper addresses these gaps. We first explore the relationships between Clarke directional derivatives of interval mappings and their endpoint functions. The Clarke subdifferential of interval mappings is defined by means of directional derivatives and interval order relations, with its fundamental properties established. A new generalized convexity concept for interval mappings is further introduced based on the proposed Clarke subdifferential. Finally, sufficient conditions for efficient solutions to interval nonsmooth optimization are derived under the new generalized convexity framework.