Proximal point methods in Banach spaces without monotonicity
Rolando Gárciga Otero | Iusem, Alfredo
hypomonotone operator | proximal point algorithm | hybrid proximal-extragradient algorithm
We introduce the concept of hypomonotone point-to-set operator in Banach spaces, with respect to a regularizing function. This notion coincides with the one given by Rockefellar and Wets in Hilbert spaces, when the regularizing function is the square of the norm. We study the associated proximal mapping, which leads to hybrid proximal-extragradient and proximal-projection methods for nonmonotone operators in reflexive Banach spaces. These methods allow for inexact solution of the proximal subproblems with relative error criteria. We consider then the notion of local hypomonotonicity, and propose localized versions of the algorithms, which are locally convergent.