Comparison with the Multidimensional Optimization

Convergence to the solution of a nonlinear equation system is often checked by means of an appropriate merit function, such as the Euclidean norm of the residuals. 

therefore, reasonable to think that the problem of solving nonlinear systems can be tackled by a minimization program applied to a merit function of this kind. Unfortunately, multimodality issues arise if this is done and many additional minima, to which the real solution does not correspond, are introduced, transforming the original problem of the nonlinear system into a multidimensional multimodal optimization problem. 

Freudenstein and Roth s problem is a good illustration (Fletcher, 1970)  

It is worth stressing that there are significant differences between the solution of nonlinear systems and multidimensional optimization. 

A solution (of local minima) is always found in optimization problems. 

---