Abstract
A novel filled function method is suggested for solving box-constrained systems of nonlinear equations. Firstly, the original problem is converted into an equivalent global optimization problem. Subsequently, a novel filled function with one parameter is proposed for solving the converted global optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed novel filled function for solving systems of nonlinear equations is presented. The objective function value can be reduced by quarter in each iteration of our algorithm. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.
Highlights
Systems of nonlinear equations arise in myriad applications, for example, in engineering, physics, mechanics, applied mathematics and sciences; see [1] for a more detailed description.In this paper, we consider the following box-constrained systems of nonlinear equations (for short, (SNE)):F (x) = 0, x ∈ X, (SNE)where the mapping F : Rn → Rm is continuous, x ⊂ Rn is a box.Generally, systems of nonlinear equations are very difficult to solve directly
It is easy to see that the objective function satisfies f(x) ≥ 0 and global optimal solutions of problem (OP) with the zero objective function value corresponding to solutions of (SNE)
In this paper, we use an efficient filled function method to solve the corresponding optimization problem, and the local minimizer of the filled function is always obtained in the interior and is always good point
Summary
Systems of nonlinear equations arise in myriad applications, for example, in engineering, physics, mechanics, applied mathematics and sciences; see [1] for a more detailed description. The traditional optimization-based methods for solving (SNE) are often stuck at a stationary point or a local minimizer of the corresponding optimization problem, which is not necessarily a solution of the original system. In this paper, we use an efficient filled function method to solve the corresponding optimization problem, and the local minimizer of the filled function is always obtained in the interior and is always good point. We propose a new filled function method, which can ensure that the proposed function is an efficient filled function and the local minimizer of the new filled function on a given box set is a better point and the primal problem’s objective value at this better point can be reduced by quarter in each iteration of our filled function algorithm.
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