Approximate first-order primal-dual algorithms for saddle point problems

We propose two approximate versions of the first-order primal-dual algorithm (PDA) to solve a class convex-concave saddle point problems. The introduced criteria are easy implement in sense that they only involve subgradient certain function at current iterate. first PDA solves both subproblems inexactly and adopts absolute error criteria, which based on non-negative summable sequences. Assuming one can be solved exactly, second other subproblem approximately relative criterion. criterion involves single parameter range $[0, 1)$, makes method more applicable. For versions, we establish global convergence $O(1/N)$ rate measured by iteration complexity, where $N$ counts number iterations. inexact with show accelerated $O(1/N^2)$ linear under assumptions part underlying functions strongly convex, respectively. Then, prove these also extended general Finally, perform some numerical experiments sparse recovery image processing results demonstrate feasibility superiority proposed methods.