Abstract
The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHG’s convergence was established only under some restrictive conditions on its step sizes. In this paper, we revisit PDHG’s convergence in the context of a saddle-point problem and try to better understand how to choose its step sizes. More specifically, we show by an extremely simple example that PDHG is not necessarily convergent even when the step sizes are fixed as tiny constants. We then show that PDHG with constant step sizes is indeed convergent if one of the functions of the saddle-point problem is strongly convex, a condition that does hold for some variational models in imaging. With this additional condition, we also establish a worst-case convergence rate measured by the iteration complexity for PDHG with constant step sizes.
Original language | English |
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Pages (from-to) | 2526-2537 |
Number of pages | 12 |
Journal | SIAM Journal on Imaging Sciences |
Volume | 7 |
Issue number | 4 |
DOIs | |
Publication status | Published - 3 Dec 2014 |
Scopus Subject Areas
- Mathematics(all)
- Applied Mathematics
User-Defined Keywords
- Convergence rate
- Convex optimization
- Image restoration
- Primal-dual hybrid gradient algorithm
- Saddle-point problem
- Total variation