Bounded Constrained Semismooth System Of Non-monotone Projected Gradient Trust Region Methods

Optimization theroy and method, which focuses on how to find the optimal solution among many feasible plans, is a very popular and useful subject. It discusses the property of the optimal solution, builds the algorithm and researches the characteristics of them. Optimization technology is widely applied in many fields such as defense, industrial production, transportation, finance, economic plan, engineering design, production management.Currently, there were many articles which researched the solutions of the bound-constrained semismooth systems. However, the methods for solving the bound-constrained semismooth under-determined systems were very limited. Actually, the applications of the underdetermined case are wider in practice. Therefore, this paper proposed two methods for solving the bound-constrained semismooth underdetermined systems, including non-monotone gradient projection trust-region strategy and non-monotone Levenberg-Marquardt projection trust-region strategy under local er-ror bound conditions.Firstly, a gradient projection trust-region strategy with non-monotone line search technique is modified and extended for solving the underdetermined system subject to simple bounds. In the (?)∞-norm trust-region problem, we establish a trust-region subproblem and get a search direc-tion for iterating. A trial step is computed by projecting a semismooth Gauss-Newton-like step onto the feasible set. Closing to a BD-regular solution, the trust-region strategy turns into the projected Gauss-Newton method. Global and local convergence properties are obtained under suitable assumptions. It can be said that our proposed algorithm widens the practice range of the non-monotone gradient projection trust-region strategy.Simultaneously, considering the disadvantage of the Gauss-Newton method, we combine the Levenberg-Marquardt method with the non-monotone gradient projection trust-region strategy to solve the bound-constrained semismooth underdetermined systems. And the proofs of the global convergence and local convergence rate are discussed to show the feasibility and effectiveness of the proposed algorithm. Then we also prove that this algorithm has a quadratic rate of conver-gence under the local error bound instead of the nonsingularity of Jacobian at a solution.Besides, this paper provides the numerical experiments by the application of the Matlab soft-ware in order to explain feasibility and effectiveness of the algorithm.The paper consists of four parts. In Chapter 1, we briefly review the basic concept of the opti-mization. In Chapter 2, we present the algorithm of non-monotone gradient projection trust-region method for solving bound-constrained semismooth underdetermined systems and the theoretical analysis of the algorithm. Global convergence and local super linear convergence rate of the proposed algorithm are established on some reasonable conditions. Furthermore, we give some relative numerical results which indicate the effectiveness of the algorithm by applying the math-ematical software Matlab. In Chapter 3, combining the Levenberg-Marquardt method with the non-monotone gradient projection trust-region strategy, we design a new algorithm to avoid the disadvantage of the Gauss-Newton Method. And the relative proofs of convergence for the con-strained semismooth system are given to show the feasibility and effectiveness of the proposed algorithm. Finally, some conclusions and the further research directions follow in Chapter 4.