I am trying to find the minimizer of the function

$\displaystyle \left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2$
s.t. $\displaystyle \mu _i \geq 0$ , $\displaystyle \mu _i = 0$ if $\displaystyle x_i > 0$

We use the function $\displaystyle \phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}$
we have that $\displaystyle \mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0$
So we can actually solve the problem

Minimize $\displaystyle \left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + \left\| \phi (\mu ) \right\|^2 $
s.t. $\displaystyle \lambda , \mu \geq 0$
Now my reasoning is, by letting $\displaystyle g= \nabla f(x) + \lambda ^T \nabla h(x) $ , the problem becomes:

Minimize $\displaystyle (g+ \mu ) ^2 + \phi (\mu ) ^2 $ i.e. Minimize $\displaystyle (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 $
or Minimize $\displaystyle (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 \quad or \quad x^2 \right\} $

Now, I think I should first find the critical points for the function $\displaystyle (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 \quad or \quad x^2 \right\} $ . But, should I consider this function as a function of $\displaystyle \mu $ and $\displaystyle g$, or as a function of $\displaystyle \mu $ and $\displaystyle x $?

Another way of thinking about this problem. If I use the 'Fischer-Burmeister' function which is:

I am trying to find the minimizer of the function

$\displaystyle \left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2$
s.t. $\displaystyle \mu _i \geq 0$ , $\displaystyle \mu _i = 0$ if $\displaystyle x_i > 0$

We use the function $\displaystyle \phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}$
we have that $\displaystyle \mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0$

So we can actually solve the problem

Minimize $\displaystyle \left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + [itex]\left\| \phi (\mu ) \right\|^2$
s.t. $\displaystyle \lambda , \mu \geq 0$
Now my reasoning is, by letting $\displaystyle g= \nabla f(x) + \lambda ^T \nabla h(x) $ , the problem becomes:

Minimize $\displaystyle (g+ \mu ) ^2 + \phi (\mu ) ^2$ i.e. Minimize $\displaystyle (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 $
or Minimize$\displaystyle (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\} $

Now, I think I should first find the critical points for the function $\displaystyle (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\}$ . But, should I consider this function as a function of $\displaystyle \mu $ and [itex]g[/TEX], or as a function of $\displaystyle \mu $ and $\displaystyle x $?

Another way of thinking about this problem. We can use the 'Fischer-Burmeister' function which is:

$\displaystyle \Phi (\mu , x ) = \mu + x - \sqrt{\mu ^2 + x^2}$ instead of the function $\displaystyle \phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}$ because for the 'Fischer-Burmeister' function,
$\displaystyle \Phi (\mu , x ) = 0 \Leftrightarrow \mu x =0 $ just like for the previous function $\displaystyle \phi _i (\mu )$ .

Now, the problem would be to

Minimize $\displaystyle (g+ \mu ) ^2 + \Phi (\mu ) ^2 $ .

Again, how should I go about finding this minimum?