Outer semicontinuity of subdifferential and normal cone mappings

Abstract

In finite dimensions, the outer semicontinuity of a set-valued mapping is equivalent to the closedness of its graph. In this article, we study the outer semicontinuity of set-valued mappings in connection with their convexifications and linearizations in finite and infinite dimensions. The results are specified to the case where the mappings involved are given by subdifferentials of extended real-valued functions or normal cones to sets. Our developments are important for applications to second-order calculus in variational analysis in which the outer semicontinuity plays a crucial role.

Keywords:

• variational analysis
• outer semicontinuity
• generalized differentiation
• subdifferential
• normal cone

AMS Subject Classifications:

• Primary: 49J52, 49J53
• Secondary: 46B20, 58C20, 90C31

Acknowledgements

The authors thank Professor Boris Mordukhovich for his valuable guidance and support.