Abstract
The nonlinear conjugate gradient (CG) method is a widely used approach for solving large-scale optimization problems in many fields, such as physics, engineering, economics, and design. The efficiency of this method is mainly attributable to its global convergence properties and low memory requirement. In this paper, a new conjugate gradient coefficient is proposed based on the Aini-Rivaie-Mustafa (ARM) method. Furthermore, the proposed method is proved globally convergent under exact line search. This is supported by the results of the numerical tests. The numerical performance of the new CG method better than other related and more efficient compared with previous CG methods.
Highlights
IntroductionAn unconstrained minimization problem has the following form: min f (x), x Rn (1)
An unconstrained minimization problem has the following form: min f (x), x Rn (1)where f : Rn → R is a continuously differentiable function The iterative formula commonly used for solving (1) is given as follows: xk+1 = xk + kdk, k = 0,1, 2,3.... (2)k are Polak-Ribiere-Polyak (PRP) [1, 2], Fletcher and Reeves (FR) [3], Wei et al [4], the ‘Aini-Rivaie-Mustafa (ARM) method [5], Hestenes and Stiefel (HS) [6], Conjugate Descent (CD) by Fletcher [7] and Dai-Yuan (DY) [8]
By using Lemma A, we can get the following convergence theorem of the MMR conjugate gradient (CG) method
Summary
An unconstrained minimization problem has the following form: min f (x), x Rn (1). K are Polak-Ribiere-Polyak (PRP) [1, 2], Fletcher and Reeves (FR) [3], Wei et al [4], the ‘Aini-Rivaie-Mustafa (ARM) method [5], Hestenes and Stiefel (HS) [6], Conjugate Descent (CD) by Fletcher [7] and Dai-Yuan (DY) [8]. Their formulas are given as follows : PRP K. where xk is the current iteration point, and k 0 is the stepsize obtained using the exact line search formula : HS K gTk yk −1 dkT−1yk −1.
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