Generalized hessian for C{sup 1+} functions in infinite dimensional normed spaces

Conference · Fri Dec 30 23:00:00 EST 1994

OSTI ID:36357

Pales, Z ^[1]

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1. Michigan State Univ., Ann Arbor, MI (United States)

Let X be a normed space with dual X* and let D {contained_in} X be a nonempty open subset of X. For a Lipschitz function f : D {yields} R, we introduce the generalized second-order directional derivative f{sup 00} as f{sup 00}(x{sub 0}; u) : = lim sup {sub {epsilon}}{sup 2}{sup {partial_derivative}}f(x + {epsilon}u) - f(x) - {epsilon}f{sup 0}(x; u) for x{sub 0} {element_of} X. For a C{sup 1} function f : D {yields} R we define f{prime}{sup 0}(x{sub 0}; u, v) : = limp sup [{del}f(x + {epsilon}u) - {del}f(x), v] for x{sub 0} {element_of} D; u, v {element_of} X. These two concepts have been used in deriving second order conditions for nonsooth problems where the data is respectively Lipschitz and C{sup 1+} (C{sup 1} with Lipshitz gradient). To compare those results we need to establish the connections between these two notions.

• OSTI ID: 36357
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
MATRICES
NONLINEAR PROBLEMS
NONLINEAR PROGRAMMING
NUMERICAL SOLUTION
PARAMETRIC ANALYSIS