Differences of convex compacta and metric spaces of convex compacta with applications: A survey.

*(English)*Zbl 1002.49022
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, 263-296 (2000).

The authors give a nice survey about the different definitions of algebraic set differences. It is pointed out that the most definitions base essentially on the Minkowski duality, i.e., on the (bijective) correspondence \(\varphi\) between the family \(Y_n\) of all convex compact sets of \(\mathbb{R}^n\) and the family \(P_n\) of all sublinear functions \(p: \mathbb{R}^n\to \mathbb{R}\).

Let \(A,B\in Y_n\) and \(p_A,p_B\in P_n\) be the associated support functions. Since the difference \(p_A- p_B\) is not convex in general, a set difference between \(A\) and \(B\) can be expressed by \[ A\ominus B= \varphi^{-1}(C(p_A- p_B)), \] where \(C(q)\) is a suitable convexification of the positively homogeneous function \(q: \mathbb{R}^n\to \mathbb{R}\). So the special set differences depend on the choice of the special convexification of \(p_A- p_B\).

In the first part of the paper, some well-known set differences are discussed and compared, especially:

\(*\) the difference using the metric projection \(C(q)= \text{Pr}_\infty(q)\) which is the solution of the optimization problem \[ \max\{q(x)- p(x)\mid\|x\|\leq 1\}\to\min,\quad p\in P_n. \] \(*\) the \(*\)-difference where \(C(q)\) is the greatest convex minorant of \(q\). By this we get the well-known representation \[ A\ominus B= \{x\mid B+ x\subset A\}. \] \(*\) the Demyanov difference where \(C(q)\) is the Clarke upper derivative of \(q\). Here we have the representation by the Clarke subdifferential according to \[ A\ominus B= \partial_{Cl}(p_A- p_B)(0). \] \(*\) the exposed difference defined by \[ A\ominus B= \{\nabla p_A(u)- \nabla p_B(u)\mid u\in T_A- T_B\}, \] where \(T_A\) and \(T_B\) are the sets of points where \(p_A\) and \(p_B\) are differentiable, i.e., where the faces \(A(u)\) and \(B(u)\) are singletons (hence exposed points).

\(*\) the quasidifferential according to \[ A\ominus B= \text{cl co}\bigcup \{A(u)- B(u)\mid u\in\mathbb{R}^n, u\neq 0\}. \] In the second part of the paper, the authors present some applications of set-differences in nonsmooth analysis, especially regarding the representation of generalized subdifferentials of DC and quasidifferentiable functions, the approximation of linear set-valued mappings and the construction of suitable metrics (the Demyanov metric and the Bartels-Pallaschke metric) in the space \(Y_n\). Convergence, continuity and differentiability properties of polyhedral-valued mappings with respect to these metrics in comparison with the Hausdorff metric are pointed out.

For the entire collection see [Zbl 0949.00047].

Let \(A,B\in Y_n\) and \(p_A,p_B\in P_n\) be the associated support functions. Since the difference \(p_A- p_B\) is not convex in general, a set difference between \(A\) and \(B\) can be expressed by \[ A\ominus B= \varphi^{-1}(C(p_A- p_B)), \] where \(C(q)\) is a suitable convexification of the positively homogeneous function \(q: \mathbb{R}^n\to \mathbb{R}\). So the special set differences depend on the choice of the special convexification of \(p_A- p_B\).

In the first part of the paper, some well-known set differences are discussed and compared, especially:

\(*\) the difference using the metric projection \(C(q)= \text{Pr}_\infty(q)\) which is the solution of the optimization problem \[ \max\{q(x)- p(x)\mid\|x\|\leq 1\}\to\min,\quad p\in P_n. \] \(*\) the \(*\)-difference where \(C(q)\) is the greatest convex minorant of \(q\). By this we get the well-known representation \[ A\ominus B= \{x\mid B+ x\subset A\}. \] \(*\) the Demyanov difference where \(C(q)\) is the Clarke upper derivative of \(q\). Here we have the representation by the Clarke subdifferential according to \[ A\ominus B= \partial_{Cl}(p_A- p_B)(0). \] \(*\) the exposed difference defined by \[ A\ominus B= \{\nabla p_A(u)- \nabla p_B(u)\mid u\in T_A- T_B\}, \] where \(T_A\) and \(T_B\) are the sets of points where \(p_A\) and \(p_B\) are differentiable, i.e., where the faces \(A(u)\) and \(B(u)\) are singletons (hence exposed points).

\(*\) the quasidifferential according to \[ A\ominus B= \text{cl co}\bigcup \{A(u)- B(u)\mid u\in\mathbb{R}^n, u\neq 0\}. \] In the second part of the paper, the authors present some applications of set-differences in nonsmooth analysis, especially regarding the representation of generalized subdifferentials of DC and quasidifferentiable functions, the approximation of linear set-valued mappings and the construction of suitable metrics (the Demyanov metric and the Bartels-Pallaschke metric) in the space \(Y_n\). Convergence, continuity and differentiability properties of polyhedral-valued mappings with respect to these metrics in comparison with the Hausdorff metric are pointed out.

For the entire collection see [Zbl 0949.00047].

Reviewer: Jörg Thierfelder (Ilmenau)

##### MSC:

49J52 | Nonsmooth analysis |

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |

58C06 | Set-valued and function-space-valued mappings on manifolds |

49J53 | Set-valued and variational analysis |