A New Prediction–Correction Primal–Dual Hybrid Gradient Algorithm for Solving Convex Minimization Problems with Linear Constraints

Abstract

The primal–dual hybrid gradient (PDHG) algorithm has been applied for solving linearly constrained convex problems. However, it was shown that without some additional assumptions, convergence may fail. In this work, we propose a new competitive prediction–correction primal–dual hybrid gradient algorithm to solve this kind of problem. Under some conditions, we prove the global convergence for the proposed algorithm with the rate of O(1/T) in a nonergodic sense, and also in the ergodic sense, in terms of the objective function value gap and the constraint violation. Comparative performance analysis of our method with other related methods on some matrix completion and wavelet-based image inpainting test problems shows the outperformance of our approach, in terms of iteration number and CPU time.