Specular differentiation in normed vector spaces and its application to nonsmooth convex optimization

Abstract: This paper introduces specular differentiation, which generalizes Gâteaux and Fréchet differentiation in normed vector spaces. Fundamental theoretical properties of specular differentiation are investigated, including the Quasi-Mean Value Theorem and Quasi-Fermat's Theorem. As an application, three numerical methods using specular differentiation are devised to optimize nonsmooth convex functions in higher-dimensional Euclidean spaces. Numerical experiments demonstrate that the proposed methods are capable of minimizing non-differentiable functions that classical methods fail to minimize.