Modification of scaled steepest descent method for solving unconstrained optimization problems - Rashidah Johari
The steepest descent (SD) method is a well-known optimization method for solving unconstrained optimization problems. The conceptual computation of SD method is straightforward and regarded as a fundamental approach in developing optimization theory. The SD method has low computational cost and matrix storage requirement considering that it does not need the computation of second derivatives to be solved to compute the search direction. Despite its attractive advantages, the rate of convergence of SD method is quite slow when near the minimum point. This occurs because of the zigzag form of the steps that occur when approaching the minimum point. This is due to the direction of the SD being not well-scaled. Therefore, the focus of this research is to present a new scaled search direction of SD method for unconstrained optimization. This modification uses exact line search and is compared with the classical SD and Zubai'ah-Mustafa-Rivaie-Ismail (ZMRI) methods. The proposed method is developed by applying a new scaling parameter inspired from Andrei's approach into the search direction of ZMRI method. The proposed method is named as RRM denoted by the name of contributors which are Rashidah, Rivaie and Mamat. The efficiency of the proposed method is studied by testing it on 20 standard test functions via Maple 18 subroutine programming. Four different initial points are chosen for each test functions to study the global convergence properties of the proposed method. The initial points are assigned ranging from the point closer to the minimum point to the point furthest away. The efficiency and robustness of the proposed method are examined by plotting the numerical results into graphs of performance profile. It is shown that RRM solves 100% of the entire test problems while ZMRI and SD solve 95% and 71.5% of the test problems, respectively. RRM is the fastest solver for about 83.75% of the test problems, whereas ZMRI and SD are only at 13.75% and 8.75% of the test problems. In terms of CPU time, RRM is still the fastest which solves about 75% of the test problems with the minimum time while ZMRI and SD are only about 17.5% and 10% of the test problems, respectively. Hence, it can be concluded that the proposed method is more efficient and robust than the classical SD and ZMRI methods.