A novel nonmonotone trust region method based on the Metropolis criterion for solving unconstrained optimization

In this paper, we propose a novel nonmonotone trust region method that incorporates the Metropolis criterion to construct a new function sequence. This sequence is used to update both the trust region ratio and the iteration criterion, increasing the likelihood of accepting the current trial step and introducing randomness into the iteration process. When the current trial step is not accepted, we introduce an improved nonmonotone line search technique to continue the iteration. This approach significantly reduces the number of subproblems that need to be solved, thereby saving computational resources. The stochastic nonmonotone technique helps the algorithm avoid being trapped in the local optima, and a global convergence is guaranteed under certain conditions. Numerical experiments demonstrate that the algorithm can be more effectively applied to a broader range of problems.