Moderately Discontinuous Homotopy

We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The $k$-th MD homotopy group is a group $MDH^b_{\bullet }$ for any $b\in [1,\infty ]$ together with homomorphisms $MD\pi ^b\to MD\pi ^{b^{\prime}}$ for any $b\geq b^{\prime}$. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For $b=1$, it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty $, the $MD$-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case.

On Notations for Conic Hulls and Related Considerations on Tangent Cones

We propose two di erent notations for cones generated by a set and for convex cones generated by a set, usually denoted by a same notation. We make some remarks on the Bouligand tangent cone and on the Clarke tangent cone for star-shaped sets and for locally convex sets. We give some applications of these remarks to a di erentiable optimization problem with an abstract constraint.

Identification of Non-Transversal Motion Bifurcations of Linkages

The local analysis is an established approach to the study of singularities and mobility of linkages. The key result of such analyses is a local picture of the finite motion through a configuration. This reveals the finite mobility at that point and the tangents to smooth motion curves. It does, however, not immediately allow to distinguish between motion branches that do not intersect transversally (which is a rather uncommon situation that has only recently been discussed in the literature). The mathematical framework for such a local analysis is the kinematic tangent cone. It is shown in this paper that the constructive definition of the kinematic tangent cone already involves all information necessary to distinguish different motion branches. A computational method is derived by amending the algorithmic framework reported in previous publications.

Optimality Conditions for Group Sparse Constrained Optimization Problems

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ μ on $$\mathrm{Y}$$ Y giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ ( Y , d , μ ) is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ W 1 , 2 ( Y , d , μ ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ x ∈ Y is the tangent cone at x of $$\mathrm{Y}$$ Y . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ x ∈ Y such a cone is a $$\mathrm{CAT}(0)$$ CAT ( 0 ) space and, as such, has a Hilbert-like structure.