Improvement On Conjugate Gradient Method

Unconstrained optimization problem has a wide range of applications in real life, conjugate gradient method is a class of commonly used algorithm for solving unconstrained optimization problem. Research on conjugate gradient method has made considerable progress, but there are still shortcomings. Based on the previous scholars’study, the paper will do major work on conjugate gradient method as follows:1.A new scaled conjugate gradient method is proposed. With the exact line search, the method reduces to the Hestenes-Stiefel conjugate gradient method and can be viewed as a modification of Hestenes-Stiefel conjugate gradient method. Firstly, it is proved that the search direction dk satisfies the sufficient descent condition with a parameter, independently of the line search condition and the assumption that the objective function is convex. Secondly, the global convergence property is established with the strong Wolfe line search. Finally, numerical testing is making, numerical comparisions are given with the Polak-Ribiere-Polyak, Hestenes-Stifel, Dai-Yuan conjugate gradient method and theirs’variants.2.A little modification is made to the parameter βLSK-1of Liu-Storey conjugate gradient method, the modified algorithm is globally convergent under two kinds of inexact line search. On one hand, the modified parameter βNLSK-1is nonnegative and has the property of (*); the search direction dk has sufficient descent property with the strong Wolfe line search, independently of selection of line search parameter σ, that is σ∈(0,1); then the global convergence of algorithm is proved. On the other hand, under a Armijo type line search, the search direction dk also satisfies the sufficient descent condition; the global convergence of algorithm is established under mild conditions, that only needs the gradient function is Lipschitz continous and does not require the level set of objective function is bounded. Finally, numerical testing is making, numerical comparisions are given with the Liu-Storey conjugate gradient method and its another modified method.