I was reading through some results on google to learn simple optimization problems in MATLAB, and found a PDF where a method is described which I don't understand.
Here is a picture of it, or see PDF page 5, Part III.
Why is the solution of the system equivalent to finding the minimum of the sum of the squares of the left hand sides?
Thank you!!
For convenience, let's label the functions in the system:
\begin{eqnarray*} f(x,y) &=& 3y + 0.005y\,\sin(5x) - 4x &=& 0\\ g(x,y) &=& x^2 + y^2 - 25 &=& 0. \end{eqnarray*}
So we have: \begin{eqnarray*} \Theta(x,y) &=& [f(x,y)]^2 + [g(x,y)]^2. \end{eqnarray*}
Now, $f(x,y)=0$ is a, slightly bent, line through the origin, and $g(x,y)=0$ is a circle centred at the origin. That geometry shows that these curves intersect, so there is a solution to the system.
It's obvious that $\Theta(x,y) = 0$ if and only if $f(x,y) = 0$ and $g(x,y) = 0$. Therefore, if we can find $(x_0,y_0)$, say, such that $\Theta(x_0,y_0) = 0$ then $(x_0,y_0)$ is a solution to the system.
Because $\Theta(x,y)$ is the sum of squares, it can't be negative, but we know from the above that it can be $0$. So minimising it is the same as solving $\Theta(x,y) = 0$. In other words, minimising $\Theta(x,y)$ is the same as solving the system.
