An extension of RMIL and Hybrid conjugate gradient method with global convergence - Nur Syarafina Mohamed
Optimization is frequently used in the fields of science, engineering and business to find the best value of the variables that yield to the best value of the performance. One of the most significant methods used un olving a large scale unconstrained optimization problem is the Conju Gradient (CG) method due to its simplicity and low memory requirement. Since it was devised by Fletcher and Reeves in 1964, several studies and modifications have been ne continuously in order to improve the method. In this research, the original contribution to the body of knowledge is to propose two new coefficients (/Jk) of hybrid and classical CG method. 'Fhe aim is to improve the ability of the existing CG method in solving unco train d optimization problems whilst maintaining the simplicity of the form . These two formulas are motivated by RMIL method which proposed by Rivaie, Mamat, Ismail and Leong method. The proposed method were examined and an yzed under exact and inexact line searche . Theor ti ally, the new CG coefficient fulfill all the convergence propertie; ufficient descent, conv rgence rate and gI hal convergence under exact line arch and ufficient descent and global onvergence for inexact line search. Num rical performan v re tested under both line searches by using twenty¬one t st functions with four ndom, initial points from small t large scale problems a suggested by Andrei. Imti I point were chosen randomly ranging from the points which are far away and clo enough from solution points. The results are analyzed by MA TLAB subroutine program with workstation, Intel Core i7, 2.2 GHz tested base on number of iterations and central processing time per unit. Results comparison done by using a performan p fH in du d by Dolan and Moore. The ew classical CG coefficient i ompar to six well-kn wn CG methods; RMIL ethod (Rivaie- Mamat-Ismail-Leong), PRP (Polak-Ribiere-Polyak) method, Fletcher and Reeves) method, HS (Hestenes- tiefel) method, CD (Conjugate-Des nt) method and DY (Dai- Yuan) method while the new hybrid is compared to HJHJ (Hybrid¬Jinbao-Han-Jiang) method, HHUS (Hybrid-Hu-Storey) method, HDY (Hybrid-Dai-Yuan) method and HLSCD (Hybrid-Liu-Storey-Conjugate-Descent) method. Results show that both of the new G coefficients outperformed other methods, both in number of iterations and central processing time per unit under all line searches. The efficiency of the new coefficients is also tested on its implementation for regression analysis problem by comparing them with the Last Square method. From this implementation, results show that the new CG coefficients are on par with the Least Square method. Under inexact line search, the new hybrid method for quadratic model gives the best estimation point due to the smallest relative error obtained. Thus, all the outcomes showed that both of the new CG coefficients have superior performances.