# F(x) = 0 vs. ||F(x)||^2->min

In many areas of application, one needs to solve a nonlinear system of equations $$F(x) = 0.$$ Sometimes, the formulation $$\|F(x)\|^2 \to\min$$ is used. Clearly, every solution $\hat{x}$ of $F(x)=0$ is also a solution of the second problem; the converse is also true (if a solution exists).

The question is if one can tell a-priori which formulation is better suited for a given problem. Have people worked on this before?

## One example

Consider the function $$F(x, y) = \begin{pmatrix} x^3 - 3x y^2 - 1\\ 3 x^2 y - y^3 \end{pmatrix}.$$ It has the three roots $x_1=(1,0)$ (green in the figure below), $x_2=(-0.5,\sqrt{3}/2)$ (blue), $x_3=(-0.5,-\sqrt{3}/2)$ (red). When applying Newton's method to $F$, the starting point will determine to which of the three solutions we converge. The darker the color, the more Newton iterations were required. The typical Newton fractals appear.

When finding critial points $\nabla (\|F(x)\|^2) = 0$, again with Newton's method, the picture is a little different. Note that the point $(0,0)$ is a critical point of $\|F(x)\|^2$, but no solution of $F(x)=0$.

This highlights one possible problem with the $\min$-formulation.