Convergence Results of a New Monotone Inertial Forward–Backward Splitting Algorithm Under the Local Hölder Error Bound Condition

In this paper, we introduce a new monotone inertial Forward–Backward splitting algorithm (newMIFBS) for the convex minimization of the sum of a non-smooth function and a smooth differentiable function. The newMIFBS can overcome two negative effects caused by IFBS, i.e., the undesirable oscillations ultimately and extremely nonmonotone, which might lead to the algorithm diverges, for some special problems. We study the improved convergence rates for the objective function and the convergence of iterates under a local Hölder error bound (Local HEB) condition. Also, our study extends the previous results for IFBS under the Local HEB. Finally, we present some numerical experiments for the simplest newMIFBS (hybrid_MIFBS) to illustrate our results.