Golden ratio primal-dual algorithm with linesearch.

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex optimization problem, in which the objective function consists of the sum of two closed proper convex functions, one of which involves a composition with a linear transform. In this paper, we propose a linesearch strategy for GRPDA, which not only does not require the spectral norm of the linear transform but also allows adaptive and potentially much larger stepsizes. Within each linesearch step, only the dual variable needs to be updated, and it is thus quite cheap and does not require any extra matrix-vector multiplications for many special yet important applications, e.g., regularized least squares problem. Global convergence and ${\cal O}(1/N)$ ergodic convergence rate results measured by the primal-dual gap function are established, where $N$ denotes the iteration counter. When one of the component functions is strongly convex, faster ${\cal O}(1/N^2)$ ergodic convergence rate results are established by adaptively choosing some algorithmic parameters. Moreover, when both component functions are strongly convex, nonergodic linear converge results are established. Numerical experiments on matrix game and LASSO problems illustrate the effectiveness of the proposed linesearch strategy.