The q-G method : A q-version of the Steepest Descent method for global optimization.

Soterroni AC, Galski RL, Scarabello MC, Ramos FM - Springerplus (2015)

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Fig7: Fraction of all problems solved as function of allowable number of function evaluations
Mentions: The resolution of all 22 derivative-free methods and the q-G method over this set of ten problems, each one solved for 10 independent runs, results in a total number of 100 optimization instances per algorithm. In order to perform a qualitative comparison of the different methods over this set of problems, we calculate for each algorithm the fraction of solved problems given by the ratio between the number of successful runs and the total number of instances. This ratio is computed every 50 evaluations of the objective function. Figures 5, 6 and 7 show the fraction of multimodal, unimodal and all problems, respectively, solved by each method to reach the optimality tolerance along the iterative procedure.Fig. 5

Bottom Line: The q-gradient vector, or simply the q-gradient, is a generalization of the classical gradient vector based on the concept of Jackson's derivative from the q-calculus.Its use provides the algorithm an effective mechanism for escaping from local minima.The algorithm has three free parameters and it is implemented so that the search process gradually shifts from global exploration in the beginning to local exploitation in the end.

View Article: PubMed Central - PubMed

Affiliation: Laboratory of Computing and Applied Mathematics, National Institute for Space Research, São José dos Campos, Brazil.

ABSTRACT
In this work, the q-Gradient (q-G) method, a q-version of the Steepest Descent method, is presented. The main idea behind the q-G method is the use of the negative of the q-gradient vector of the objective function as the search direction. The q-gradient vector, or simply the q-gradient, is a generalization of the classical gradient vector based on the concept of Jackson's derivative from the q-calculus. Its use provides the algorithm an effective mechanism for escaping from local minima. The q-G method reduces to the Steepest Descent method when the parameter q tends to 1. The algorithm has three free parameters and it is implemented so that the search process gradually shifts from global exploration in the beginning to local exploitation in the end. We evaluated the q-G method on 34 test functions, and compared its performance with 34 optimization algorithms, including derivative-free algorithms and the Steepest Descent method. Our results show that the q-G method is competitive and has a great potential for solving multimodal optimization problems.

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