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I have a question regarding multivariable optimization. In particular, I have a function f(x,y,z,w) and I want to maximize f in terms of x only (with other variables treated as parameters). Also I have one more constraint: y=g(x), that is, this constraint is an equation of only y and x.

To maximize f, all I have to do is to substitude y for y=g(x), and then I $d/dx[f(x,y=g(x),z,w)]=0$, then just solve for x.

Here is the problem. I don't think I can solve for x.

So I decide to do the followings:

1) substitude $x=g^{-1}(y)$ to f.

2) f is now $f(g^{-1}(y),y,z,w)$

3)$d/dy[f(g^{-1}(y),y,z,w)]$=0, then solve for y.

4) after I obtain answer for y, then using the constraint, I could get x.

Now, do you think what I do in above 4 steps optimize f in terms of x?

kou
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1 Answers1

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For this problem, I think that there are two different approaches. You want to maximize $$F(x,y,z,w)$$ subject to the the contraint $y=g(x)$, $z$ and $w$ being parameters.

The first approach (what you did) could be to write $$F(x,y(x),z,w)$$ and compute its derivative with respect to $x$ which is a single equation in $x$. This would give $$y'(x)F'_y+F'_x=0$$

The second approach will imake use of Lagrange multipliers and this would correspond to the maximization of $$\Phi(x,y,z,w)=F(x,y,z,w)+\lambda \Big(y-g(x)\Big)$$ but this will require to solve $$\Phi'_x=\Phi'_y=\Phi'_{\lambda}=0$$.

I definitely think that the first approach is the simplest.

Claude Leibovici
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  • Do u think it is correct with my 4 steps? Because I want to optimize in terms of x, but in those 4 steps, I can taking derivative with y – kou Jul 19 '14 at 06:25
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    Honestly, I think that you are making your life more difficult than required. I added to my answer. Could you provide an example ? Cheers :) – Claude Leibovici Jul 19 '14 at 06:38