- #1

math8

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[itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2[/itex]

s.t. [itex]\mu _i \geq 0[/itex] , [itex]\mu _i = 0[/itex] if [itex]x_i > 0[/itex]

We use the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex]

we have that [itex]\mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0[/itex]

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 [/itex]

s.t. [itex]\lambda , \mu \geq 0[/itex]

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

Minimize [itex] (g+ \mu ) ^2 + \phi (\mu ) ^2 [/itex] i.e. Minimize [itex] (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 [/itex]

or Minimize [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 or \quad x^2 \right\} [/itex]

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

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

[itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2[/itex]

s.t. [itex]\mu _i \geq 0[/itex] , [itex]\mu _i = 0[/itex] if [itex]x_i > 0[/itex]

We use the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex]

we have that [itex]\mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0[/itex]

So we can actually solve the problem

Minimize [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + [itex]\left\| \phi (\mu ) \right\|^2 [/itex]

s.t. [itex]\lambda , \mu \geq 0[/itex]

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

Minimize [itex] (g+ \mu ) ^2 + \phi (\mu ) ^2 [/itex] i.e. Minimize [itex] (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 [/itex]

or Minimize [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\} [/itex]

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

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

[itex]\Phi (\mu , x ) = \mu + x - \sqrt{\mu ^2 + x^2} [/itex] instead of the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex] because for the 'Fischer-Burmeister' function,

[itex]\Phi (\mu , x ) = 0 \Leftrightarrow \mu x =0 [/itex] just like for the previous function [itex]\phi _i (\mu )[/itex] .

Now, the problem would be to

Minimize [itex] (g+ \mu ) ^2 + \Phi (\mu ) ^2 [/itex] .

Again, how should I go about finding this minimum?