Further Studies And Improvements On Conjugate-Gradient-Type Methods

Due to simplicity, low storage and high efficiency, nonlinear conjugate gradientmethod is a powerful tool for solving large scale unconstrained optimization problems,and it has been applied to many fields. However, in order to improve efficiency andachieve good theoretical results, we should turn our attention to conjugate-gradient-typemethods. Conjugate-gradient-type methods are a class of methods whose iterativeformulas are similar to those of the classical ones, yet they can not reduce to the laterwhen line search is exact and the objective function is strictly convex quadratic. Thefollowing two new methods are proposed based on the studies of conjugate gradientmethods from both home and abroad:1. A new general form of conjugate gradient methods is derived. Nice convergenceproperties can be obtained under mild conditions while satisfying the sufficient descentcondition without line searches. Based on this new method, convergence results onPolak-Ribiere-Polyak，Hestenes-Stiefel，Liu-Storey，Dai-Yuan and Conjugate-Descentmethods are established. By testing lots of unconstrained optimization problems fromCUTEr library, the efficiency of the proposed method is shown.2. A special search direction d_k is derived according to the modified conjugacycondition proposed by Dai and Liao. Powell’s restart criterion is made use of toovercome the main difficulty of the DL method that it can not guarantee the descentproperty. Good theoretical results and numerical performance are gained finally.