Point-Based Set-Valued Approximations, C -Differential Operators and Applications

Abstract

In this article, we investigate the point-based set-valued approximation [H. Xu (1999). Set-valued approximation and Newton's methods. Mathematical Programming , 84 , 401-420.] and the C -differential operator [L. Qi (1996). C -Differential operators, C -differentiability and generalized Newton methods, Applied Mathematics Report, AMR96/5, School of Mathematics, The University of New South Wales, Sydney, Australia.] of a vector valued function. Several questions about C -differential operators posed in [L. Qi (1996). C -Differential operators, C -differentiability and generalized Newton methods, Applied Mathematics Report, AMR96/5, School of Mathematics, The University of New South Wales, Sydney, Australia.] are answered. Further, we discuss the application of C -differential operator in nonsmooth optimization. We derive optimality conditions and establish a mean value theorem in terms of convex C -differential operators. Finally, we propose a conceptual algorithm based on the convex C -differential operators.