Douglas-Rachford Algorithm for The Generalized DC Programming in Hilbert Spaces

Chih-Sheng Chuang

Department of Applied Mathematics

National Chiayi University

cschuang@mail.ncyu.edu.tw, cschuang1977@gmail.com

    In this paper, we first study the generalized DC programming in real Hilbert space:

`(\mathbf{GDCP})\text{ argmin}_{x\in H}\{f(x)=\varphi(x)+g(x)-h(x)\},`

where `H` is a real Hilbert space, `\varphi,g:H\rightarrow (-\infty,\infty]` are proper, strongly convex, and lower semicontinuous functions, and `h:H\rightarrow (-\infty,\infty]` is a Fr\'{e}chet differentiable with the gradient `\nabla f(x)`. Next, we give the Douglas-Rachford algorithm and Peaceman-Rachford algorithm to study the generalized DC programming. Next, we also study the split generalized DC programming by proposing a split Douglas-Rachford algorithm.

Keyword: Douglas-Rachford algorithm, Peaceman-Rachford algorithm, DC programming, strongly convexity, strongly monotonicity