In this paper, aiming at the nonlinear equations, a new two-step Levenberg–Marquardt method was proposed. We presented a new Levenberg–Marquardt parameter to obtain the trial step. A new modified Metropolis criterion was used to adjust the upper bound of the approximate step. The convergence of the method was analyzed under the H $\ddot{\mathrm{o}}$lderian local error bound condition and the H $\ddot{}o$lderian continuity of the Jacobian. Numerical experiments showed that the new algorithm is effective and competitive in the numbers of functions, Jacobian evaluations and iterations.

In this work, we proposed a new trust region method for solving large-scale unconstrained optimization problems. The trust region subproblem with a simple form was constructed based on new weak secant equations, which utilized both gradient and function values and available information from the three most recent points. A modified Metropolis criterion was used to determine whether to accept the trial step, and an adaptive strategy was used to update the trust region radius. The global convergence and locally superlinearly convergence of the new algorithm were established under appropriate conditions. Numerical experiments showed that the proposed algorithm was effective.

In this paper, we propose a modified Levenberg–Marquardt (LM) method with a nonmonotone technique for solving nonlinear equations. Under the $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}$ continuity and the $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}$ local error bounds conditions, which are weaker than the local error bounds and the Lipschitz continuity, the global convergence and local convergence of the algorithm are proved. Numerical experiments also show that the algorithm is effective.

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.