The New Cone Model Trust Region Algorithm With Backtracking Line Search And Diminuendo Filtering Technology

For solving unconstrained optimization problems, the trust region algorithmis very popular with the mature theoretical properties and well numerical. Themain idea of the trust region method is that, we select a model function first, anddefine a reliable region in the near of the current iteration point in each iteration,and we assume that the model function is a sufficient approximation to the targetfunction in this area. Then we get an approximate minimizer as the next iterationfor calculating the model function within the trust region, and this process isrepeated until convergence.The traditional trust region method often used quadratic models for solvingthis problem. However, for strongly non-quadratic form and more dramaticchange of curvature function, the quadratic models are often not a goodapproximation of the objective function. Davidon was first proposed the conicfunction in1980, and show that the conic function with approximation of theobjective function is better when we solve the unconstrained optimizationproblems. The Conic model used more information from the previous pointsthan the quadratic models. Especially, the new trust region algorithm of conicmodel is proposed in2005by Ni-Qin which abolished on the constraint of thetrust region radius and the horizontal vector.It overcomes the disadvantages ofunbounded possibility with the conic function, and conic model method researchhas be opened up a broad space for development.This article mainly studies the main idea of algorithm and improvement. In this paper I introduce the development of more sophisticated technology of thequadratic models into the new conic model, form a new class of algorithms, andthe numerical experiment shows convergence and feasibility the algorithm andthe numerical test for the detection of the convergence of the algorithm and thefeasibility. The work done mainly concentrated in the third to the fifth chapter.In the third chapter, a trust region algorithm with backtracking line search isproposed based on the new conic model. And the adjustment of the trust regionradius used the self-adaptive.In the forth chapter, a weak filter technology isintroduced into the trust region algorithm based on the new conic model. That isto say I introduced a weak function into the traditional algorithm whit filtertechnology. Extended Rosenbrock function and Extended Powell singularfunction are tested and get good results. In the fifth chapter, a multidimensionalalgorithm with backtracking line search and weak filter trust region algorithm isproposed based on the new conic model. Finally, the global convergence of thenew algorithm is proved, and get better results.