A new filled function for global minimization and system of nonlinear equations

Filled function methods have been considered as an effective approach for solving global optimization problems. However, most filled functions have the drawbacks of discontinuity, non-differentiability, and they could be sensitive to parameters and have exponential or logarithmic terms which may reduce their efficiency. In this paper, we propose a continuously differentiable filled function without parameters and exponential/logarithmic terms to overcome these problems. The continuous differentiability of the presented filled function makes its minimization an easy process and allows using efficient indirect local search methods. The proposed filled function has no parameters to adjust. Adjustment of the parameters is not an easy task, since the parameters may take different values for different problems. Moreover, the new filled function is numerically stable, since there are no exponential or logarithmic terms. Theoretical features of the considered filled function are investigated and a new algorithm for unconstrained global optimization problems is designed. The numerical results show that this method can successfully be used to solve global optimization problems, with a large number of variables. Furthermore, we extend the proposed filled function method to solve systems of nonlinear equations. Finally, some test problems for systems of nonlinear equations are reported, with satisfactory numerical results.