Two Convergent Primal–Dual Hybrid Gradient Type Methods for Convex Programming with Linear Constraints

As an effective tool for convex programming, the primal–dual hybrid gradient (PDHG) method has been widely applied in science and engineering computing field. However, the PDHG method may be divergent without further assumption. Based on the projection and contraction methods for monotone variational inequalities, this paper presents two convergent PDHG-type (C-PDHG) methods for the convex programming with linear constraints, whose proximal parameters r and s only need to satisfy r 0, s 0, rs 1 4 AT A and r 0, s 0, respectively. The global convergence of the two new C-PDHG methods is proved. Furthermore, the convergence rate for linearly equality constrained programming is also studied. Finally, some numerical results on the linear support vector machine and the image reconstruction are reported to show the efficiency of the two C-PDHG methods.