A new primal-dual hybrid gradient scheme for solving minimax problems with nonlinear term

Primal-dual hybrid gradient (PDHG) methods are popular for solving minimax problems. The proximal terms in the corresponding subproblems play an important role in the convergence analysis and for numerical performance of PDHG methods. However, it is observed that the function values generated by some PDHG algorithms might suffer from intense oscillation as the iteration progresses. To address the drawback, we take advantage of an inertial point to exploit a new proximal term, construct a new quadratic approximation for the nonlinear term in the minimax problem, and present a new primal-dual hybrid gradient algorithm for solving minimax problems with nonlinear terms. The new proximal term is different from other commonly used proximal terms and is used in the x-subproblem of the proposed algorithm. The quadratic approximation is used to replace the common linear approximation in the subproblem of the proposed algorithm to accelerate the proposed method. The local convergence of the proposed algorithm is established under mild assumptions. Numerical experiments on two examples confirm the compelling numerical performance of the proposed method.