How to Determine the Minimizer for a Complex Optimization Problem?

[math8]

I am trying to find the minimizer of the function

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

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

So we can actually solve the problem

Minimize $\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + \left\| \phi (\mu ) \right\|^2$
s.t. $\lambda , \mu \geq 0$

Now my reasoning is, by letting $g= \nabla f(x) + \lambda ^T \nabla h(x)$ , the problem becomes:

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

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

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

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

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

So we can actually solve the problem

Minimize [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + $\left\| \phi (\mu ) \right\|^2$
s.t. $\lambda , \mu \geq 0$

Now my reasoning is, by letting $g= \nabla f(x) + \lambda ^T \nabla h(x)$ , the problem becomes:

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

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

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

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

Now, the problem would be to

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

Again, how should I go about finding this minimum?

 

[kurt.physics]

Is this a function of \mu and g, or \mu and x? The answer to your question is that the problem should be minimized as a function of \mu and g. The Fischer-Burmeister function does not change the way the problem is minimized. The problem is still to find the critical points of the function (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\}. The only difference is that now you are using the Fischer-Burmeister function instead of the min function.