# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Primal-dual gradient structured functions: second-order results; links to epi-derivatives and partly smooth functions. (English) Zbl 1036.90067
The paper studies second order expansions for the recently introduced class of nonsmooth functions with primal-dual gradient structure. For this class of lower semi-continuous and not necessarily convex functions it is possible to explicitly give a basis for the subspace ${\cal V}$ parallel to the Clarke subdifferential at some point. Relative to its orthogonal subspace ${\cal U}={\cal V}^\perp$ the function appears to be smooth, and it is actually possible to find smooth trajectories tangent to ${\cal U}$ along which the function is $C^2$. Along with this smooth restriction a smooth multiplier function can be defined. Having these two smooth objects at hand, a $C^2$ Lagrangian is defined which leads to a second order expansion of the nonsmooth function along the subspace ${\cal U}$. Explicit expressions for the first and second order derivatives are given. Connections between second order epi-derivatives and ${\cal U}$-Hessians are made, and expressions for manifold restricted Hessians are given for partly smooth functions. A number of illuminating examples accompany the results of the article.

##### MSC:
 90C31 Sensitivity, stability, parametric optimization 49J52 Nonsmooth analysis (other weak concepts of optimality) 90C46 Optimality conditions, duality
Full Text: