A Self-Correcting Geometry Wedge Trust Region Method In Derivative-Free Optimization

In this paper, we study the new derivative-free method for solving unconstrained optimization problems and nonlinear complementarity problems.Derivative-free optimization is the method only using the function value and without any derivatives. So far, several efficient methods have been proposed for solving unconstrained optimization without derivatives. Here we consider trust re-gion methods based on interpolation models. The model function, i.e., the objective function of the subproblem, in each iteration is formed by interpolation and needs to satisfy some conditions to get a good iterative point. How to build appropriate models becomes a puzzle, so far there are three main methods:the geometry-improvement step, the wedge trust region method and the self-correcting geometry process.We propose a new self-correcting geometry process and combine the wedge trust region method, so we get a new derivative-free method for unconstrained problems. The new self-correcting geometry process adopts different updating strategies of in-terpolation point set and trust region radius to accelerate the convergence. We prove the new process also has the self-correcting property. What’s more, it can take the po-sition factor of the interpolation points and avoid the defects of only considering the position factor to combine the wedge trust region method. Numerical experiments in-dicate that the modified method is efficient. Under the general assumptions, we prove the convergence.In Chapter4, the new method is used to solve nonlinear complementarity prob-lems, through solving the transformed problems by merit function. If the regularity condition holds, then any accumulation point generated by the algorithm is the solu-tion of the NCR Numerical experiments show that our method works relatively better than the derivative-free descent method by J.S. Chen and S.H. Pan [1]. What’s more, the regularity condition is weaker than the convergence condition in derivative-free descent method.