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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Set-valued analysis 4 (1996), S. 101-117 
    ISSN: 1572-932X
    Keywords: 49J52 ; 49K30 ; second-order subdifferentials ; Clarke's subdifferential ; twice strict differentiability ; sufficient optimality conditions ; necessary optimality conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present second-order subdifferentials of Clarke's type of C 1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ℝ n , considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C 1,1 data are obtained.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Set-valued analysis 5 (1997), S. 1-35 
    ISSN: 1572-932X
    Keywords: submonotone mappings ; subdifferentials ; differentiability ; quasi-differentiable functions ; Asplund spaces ; minimal w *-cusco mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The notions ‘submonotone’ and ‘strictly submonotone’ mapping, introduced by J. Spingarn in ℝ n , are extended in a natural way to arbitrary Banach spaces. Several results about monotone operators are proved for submonotone and strictly submonotone ones: Rockafellar's result about local boundedness of monotone operators; Kenderov's result about single-valuedness and upper-semicontinuity almost everywhere of monotone operators in Asplund spaces; minimality (w*-cusco mappings) of maximal strictly submonotone mappings, etc. It is shown that subdifferentials of various classes of nonconvex functions defined as pointwise suprema of quasi-differentiable functions possess submonotone properties. Results about generic differentiability of such functions are obtained (among them are new generalizations of an Ekeland and Lebourg's theorem). Applications are given to the properties of the distance function in a Banach space with a uniformly Gateaux differentiable norm.
    Type of Medium: Electronic Resource
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