A Quasi-Newton Primal-Dual Algorithm with Line Search

Abstract

Quasi-Newton methods refer to a class of algorithms at the interface between first and second order methods. They aim to progress as substantially as second order methods per iteration, while maintaining the computational complexity of first order methods. The approximation of second order information by first order derivatives can be expressed as adopting a variable metric, which for (limited memory) quasi-Newton methods is of type "identity ± low rank". This paper continues the effort to make these powerful methods available for non-smooth systems occurring, for example, in large scale Machine Learning applications by exploiting this special structure. We develop a line search variant of a recently introduced quasi-Newton primal-dual algorithm, which adds significant flexibility, admits larger steps per iteration, and circumvents the complicated precalculation of a certain operator norm. We prove convergence, including convergence rates, for our proposed method and outperform related algorithms in a large scale image deblurring application. Keywords: quasi-Newton • primal-dual algorithm • line search • saddlepoint problems • large scale optimization ⋆ We acknowledge funding by the ANR-DFG joint project TRINOM-DS under the number DFG OC150/5-1.