An inexact semismooth Newton-Krylov method for semilinear elliptic optimal control problem

Abstract: An inexact semismooth Newton method has been proposed for solving semi-linear elliptic optimal control problems in this paper. This method incorporates the generalized minimal residual (GMRES) method, a type of Krylov subspace method, to solve the Newton equations and utilizes nonmonotonic line search to adjust the iteration step size. The original problem is reformulated into a nonlinear equation through variational inequality principles and discretized using a second-order finite difference scheme. By leveraging slanting differentiability, the algorithm constructs semismooth Newton directions and employs GMRES method to inexactly solve the Newton equations, significantly reducing computational overhead. A dynamic nonmonotonic line search strategy is introduced to adjust stepsizes adaptively, ensuring global convergence while overcoming local stagnation. Theoretical analysis demonstrates that the algorithm achieves superlinear convergence near optimal solutions when the residual control parameter $η_k$ approaches to 0. Numerical experiments validate the method's accuracy and efficiency in solving semilinear elliptic optimal control problems, corroborating theoretical insights.