Continuous approximations, codifferentiable functions and minimization methods. (English) Zbl 0973.49013

Demyanov, V. (ed.) et al., Quasidifferentiability and related topics. Dedicated to Prof. Franco Giannessi on his 65th birthday and to Prof. Diethard Pallaschke on his 60th birthday. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 43, 361-391 (2000).
The function \(f^{\odot}: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\) is called a first-order approximation of the (nonsmooth) function \(f: \mathbb{R}^n\to \mathbb{R}\) if for all \(x\in \mathbb{R}^n\) and \(d\in\mathbb{R}^n\) \[ f(x+ d)- f(x)= f^{\odot}(x, d)+ o_x(d),\quad\text{where }\lim_{\alpha\downarrow 0} {o_x(\alpha d)\over \alpha}= 0. \] Clearly, if \(f^{\odot}(x,d)\) is positively homogeneous, then \(f^{\odot}(x,d)\) coincides with the directional derivative \(f'(x,d)\) and topological properties of \(f^{\odot}(\cdot,d)\) are closely connected with algebraic properties of \(f^{\odot}(x, \cdot)\). So the directional derivative \(f'(x,\cdot)\) is linear (resp. sublinear) if \(f'(\cdot,d)\) is continuous (resp. upper semicontinuous). Hence, for a nonsmooth function the directional derivative can not be continuous with respect to \(x\).
To get continuous first-order approximations of nonsmooth functions (which are important especially for numerical methods) the author uses approximations which are not homogeneous. According to Demyanov, the function \(f\) is called to be codifferentiable if there exists a pair of convex compact sets \(\underline df(x),\overline df(x)\subset \mathbb{R}^{n+1}\) (the codifferential of \(f\)) such that \[ f(x+ d)- f(x)= \max_{(a,u)\in \underline df(x)} [a+\langle u,d\rangle]+ \min_{(b,w)\in\overline df(x)} [b+ \langle w,d\rangle]+ o_x(d). \] Hence, codifferentiable functions admit first-order approximations \(f^{\odot}(x, d)\) which are difference convex with respect to \(d\). The class of codifferentiable functions coincides with the class of quasidifferentiable functions. However, in contrast to quasidifferentiability there exists a large class of nonsmooth functions which allow Hausdorff-continuous codifferentials.
In the paper, the author summarizes calculus rules for (continuously) codifferentiable functions and provides applications to nonsmooth optimization problems. The second part of the paper is devoted to the extension of the concept of codifferentiability to mappings defined between Banach spaces.
For the entire collection see [Zbl 0949.00047].


49J52 Nonsmooth analysis
90C26 Nonconvex programming, global optimization