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As per Lagrange theorem it says I can get maximum or minimum under some constraint.
Example:

$f(x,y)=x^2+y^2+4$ under constraint $x+y=2$. I can use Lagrange theorem for this.

But I have a problem like $f(x,y)=x^2+y^2+4$ under constraint like $x$ should be minimum.

How can I solve this any theorem or solution?

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    @EuYu: I assume the OP is just trying to give an example. A better statement might be "Minimize $f$ under the constraint $g$ is also minimized."2013-05-15

3 Answers 3

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I doubt if a general solution for multiple optimalization exists, since the goals might be just conflicting with each other. But I assume you must observed that the function $x^{2}+y^{2}+4$ has an absolute minimum at $x=0,y=0$. On the other hand if you let $x=0$, $f(0,y)=y^{2}+4$ has no maximum at all. So there is essentially only one choice.

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Use $∇f = -\lambda∇g$. $\lambda$ is called the Lagrange multiplier.

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Use Lagrange multipliers with the constraint $\partial g/\partial x=0$. This will at least constrain $g$ to be extremal. So construct an auxillary function

$F(x,y,\lambda)=f(x,y)+\lambda \frac{\partial g(x)}{\partial x}$

and minimize with $\nabla F=0$