MPRP And TMPRP Type Methods For The Optimization Problems With Nonnegative Constraints

Optimization problems have a wide range of practical background in economics,management, engineering and so on. Conjugate gradient methods are a kind of the mosteffective methods for solving optimization problems. Because the methods are simple andneed lower storage, they are particularly welcome in the solution of large scale problems.The descent of traditional conjugate gradient methods is dependent of line searchused. Recently, the research in the descent conjugate gradient methods received extensiveattention, and it has made significant progress in solving unconstrained optimizationproblems. However, so far, the research in solving constrained problems with conjugategradient methods is rare. One of the main work of this thesis is to apply the idea ofdescent conjugate gradient methods that are used to solve unconstrained optimizationproblems to solve the optimization problems with nonnegative constraints. Under weakerconditions, we establish the global convergence theorem.The monotone linear search techniques are a kind of commonly used methods inlinear search technique methods. An advantage of this kind of linear search lies in thatthe generated sequence of function values is monotonically decreasing. On the other hand,however, a monotone linear search may generate a small step-length in some cases. Asa remedy, Grippo and Zhang et al proposed two non-monotone linear search techniques.Numerical results have shown that this kind of linear search works very well. Anotherimportant work of this thesis is to introduce the non-monotone linear search techniquesto the conjugate gradient methods witch are used to solve the optimization problems withnonnegative constrained, and establish their global convergence.We also do numerical experiments to test the proposed methods, and compare theperformance of the methods with Zoutendijk method. The numerical results show thatthe method that we propose outperforms the Zoutendijk method. We also do numericalexperiments to test the methods witch use non-monotone linear search techniques, Thenumerical results show that the non-monotone methods performs very well.This thesis was supported by Chinese NSF grant(10771057).