Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
On the convergence property of the DFP algorithm
Annals of Operations Research
Efficient generalized conjugate gradient algorithms, Part 1: theory
Journal of Optimization Theory and Applications
CUTE: constrained and unconstrained testing environment
ACM Transactions on Mathematical Software (TOMS)
A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property
SIAM Journal on Optimization
A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search
SIAM Journal on Optimization
Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent
ACM Transactions on Mathematical Software (TOMS)
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Although the study of global convergence of the Polak---Ribière---Polyak (PRP), Hestenes---Stiefel (HS) and Liu---Storey (LS) conjugate gradient methods has made great progress, the convergence of these algorithms for general nonlinear functions is still erratic, not to mention under weak conditions on the objective function and weak line search rules. Besides, it is also interesting to investigate whether there exists a general method that converges under the standard Armijo line search for general nonconvex functions, since very few relevant results have been achieved. So in this paper, we present a new general form of conjugate gradient methods whose theoretical significance is attractive. With any formula β k 驴驴驴0 and under weak conditions, the proposed method satisfies the sufficient descent condition independently of the line search used and the function convexity, and its global convergence can be achieved under the standard Wolfe line search or even under the standard Armijo line search. Based on this new method, convergence results on the PRP, HS, LS, Dai---Yuan---type (DY) and Conjugate---Descent---type (CD) methods are established. Preliminary numerical results show the efficiency of the proposed methods.