Golden Ratio Primal-Dual Algorithm with Linesearch

The golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow--Hurwicz method for solving structured convex optimization problems, in which objective function consists sum two closed proper functions, one involves composition with linear transform. same as and popular (PDA) Chambolle Pock, GRPDA full-splitting sense that it does not rely on any subproblems or system equations iteratively. Compared PDA, an important feature permits larger primal dual stepsizes. However, stepsize condition requires spectral norm transform known, can be difficult to obtain some applications. Furthermore, constant stepsizes are usually overconservative practice. In this paper, we propose linesearch strategy GRPDA, only require but also allows adaptive potentially much Within each step, variable needs updated, thus quite cheap extra matrix-vector multiplications many special yet applications such regularized least-squares problem. Global iterate convergence ${\cal O}(1/N)$ ergodic rate results, measured by value gap constraint violations equivalent problem, established, where $N$ denotes iteration counter. When component functions strongly convex, faster O}(1/N^2)$ quantified measures, established adaptively choosing algorithmic parameters. Moreover, when subdifferential operators metric subregular, weaker than strong convexity, show iterates converge R-linearly unique solution. Numerical experiments matrix game LASSO problems illustrate effectiveness proposed strategy.