Banach spaces of d.c. functions and quasidifferentiable functions.

*(English)*Zbl 0678.90073Summary: In recent years, quasidifferentiable functions (q.d. functions) and functions that are representable as differences of convex functions (d.c. functions) have emerged as natural tools in the study of many nondifferentiable optimization problems. In this paper we shall be concerned with some properties of the spaces of these functions.

In the first section we introduce two definitions of quasidifferentiability. These definitions extend the concept of quasidifferentiability to functions on a non-necessarily open set. They will include as special cases both Demyanov-Rubinov’s and Shapiro’s definitions of q.d. functions.

The second section is devoted to properties such as continuity, Lipschitz property, integral representability of directionally differentiable functions. The results obtained in this section will play an important role in the study of the space of q.d. functions.

The basic results of the paper are presented in Section 3, where we prove some theorems about Banach spaces of d.c. functions and q.d. functions.

In the final section, we consider q.d. and d.c. functions on [0,1]. Due to the special structure of [0,1], these functions have a number of interesting properties.

In the first section we introduce two definitions of quasidifferentiability. These definitions extend the concept of quasidifferentiability to functions on a non-necessarily open set. They will include as special cases both Demyanov-Rubinov’s and Shapiro’s definitions of q.d. functions.

The second section is devoted to properties such as continuity, Lipschitz property, integral representability of directionally differentiable functions. The results obtained in this section will play an important role in the study of the space of q.d. functions.

The basic results of the paper are presented in Section 3, where we prove some theorems about Banach spaces of d.c. functions and q.d. functions.

In the final section, we consider q.d. and d.c. functions on [0,1]. Due to the special structure of [0,1], these functions have a number of interesting properties.