| Summary: | The conjugate gradient (CO) method is the best in iterative methods due to its simple algorithm, low memory storage, and good convergence analysis. However, a major problem of the existing CO methods is that it can be very slow on the certain type of unconstrained optimization problems. Therefore, there is needed to improve the efficiency of those methods based on the number of iterations and central processing times per unit. This research proposes two new CO methods as NRM and NRM 1 methods, based on the researcher's name (Nur Hamizah, Rivaie and Mustafa) and classified into the groups of classical and hybrid CO methods, respectively. Both approaches are tested based on the exact and inexact line search. Theoretical proofs are shown that both new CO methods pos ess the sufficient descent and global convergence properties. The efficiency of the new CO methods are studied by testing on 30 standard test problems of unconstrained optimization functions, with a total of 109 problems based on four different initial points ranging from small dimensions to large dimensions problems utilizing MatlabR2012 programming. The NRM and NRM 1 methods are compared with the existing CO methods, which are Hestenes¬Stiefel (HS), Rivaie-Mustafa-I mail-Leong (RMIL), Touati-Ahmed-Storey (TS) and Jinbao-Han-Jiang (JHJ). Comparison based on performance profile shows that both new CO methods are efficient in their cla s of CO. From the exact line search, the NRM I method shows the highest successful percentage to solve test functions in group of hybrid CO compared to TS and JHJ methods. Meanwhile in group of classical CO, the NRM and RMIL method show the same highest successful percentage to solve the test functions. From the inexact line search, the NRM 1 and NRM methods show the highest successful percentage to solve test functions compared to the TS, JHJ, HS and RMIL methods. Both new methods are able to lessen the number of iterations and central processing times per unit. Moreover, the performance of CO methods in inexact line search is better than the performance in exact line search. Besides that, the numerical result also shows that new CO methods have capability to be applied in regression analysis problem. Thus, both of the new methods show promising results to be implemented in further study.
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