Duality and subdifferential for convex functions on complete CAT(0) metric spaces

作者：Kakavandi Bijan Ahmadi; Amini Massoud^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 73(10): 3450-3455.

DOI：10.1016/j.na.2010.07.033

• Hadamard space
• Dual space
• Quasilinearization
• Subdifferential

Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete CAT(0) (Hadamard) spaces. For a Hadamard space X, its dual is a metric space X* which strictly separates non-empty, disjoint, convex closed subsets of X, provided that one of them is compact. If f : X -> 1 (-infinity, + infinity] is a proper, lower semicontinuous, convex function, then the subdifferential partial derivative f : X reversible arrow X* is defined as a multivalued monotone operator such that, for any y is an element of X there exists some x is an element of X with (xy) over right arrow is an element of partial derivative f(x). When X is a Hilbert space, it is a classical fact that R(I + partial derivative f) = X. Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential partial derivative(epsilon)f(x) is non-empty, for any epsilon > 0 and any x in efficient domain of f. Our results generalize duality and subdifferential of convex functions in Hilbert spaces.

• 出版日期2010-11-15