On morse theory for piecewise smooth functions

Abstract

ABSTRACT. Lower level sets of continuous selections of C2-functioas defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of [11] are studied. It is shown that the lower level set is homotopy equivalent to the join of the lower level sets of the smooth and the nonsmooth part, respectively, of the corresponding normal form. Some generalized Morse inequalities are deduced from this result.

Topological Aspects of Nonsmooth Optimization

“…with local maximizers [61]. Finally, we refer to [2,6,90] for the results on Morse theory for piecewise smooth functions.…”

“…Clearly, question (2) means that some desirable properties should hold at regular points, such as metric or topological properties. Section 6.2 will be devoted to this issue.…”

Isotopic Implicit Surface Meshing

“…But the gradient of g at any point p of U 1 where it is defined is a convex combination of ∇f (p) and ∇f (p) : it thus makes a positive dot product with ∇ f (p). By the result of [1] which we mentioned when we stated lemma 2, this implies that the index of p is 0. We thus proved the announced claim.…”

“…The index of a vertex v is defined to be 1 minus the Euler characteristic of Lk − (v) [2]. In particular, regular points all have 1 By simplex we mean a closed cell of T of any dimension. 2 A complex is collapsible if it collapses to a point.…”

Tangentially ds functions

“…However, these sets are no more weakly compact. It follows that we have to revisit the results obtained by Pallaschke and Urbanski and their co-authors ( [1,2,7,10,20,[25][26][27][28][29][30][31][32][33][34]) and the ones by Demyanov, Rubinov and their co-authors ( [8][9][10][11][12][13][14][15][16][17]). In view of the fact that we have to use operations in R, we adopt the rules devised by Moreau [24].…”