0
$\begingroup$

Help me to find general solution of the following optimization problem: minimize $F(x,y)=(x^2-x_0)^2+y^2 $ subject to $y^2+x^2=Q$ where $x, y \in R$.

Thanks in advice!

  • 0
    This is the right-typed question.2011-11-08

1 Answers 1

1

Edit: Note that the problem reduces to minimizing $(a-x_0)^2+b$ subject to $a+b=Q$ which in turn reduces to minimizing $(Q-b-x_0)^2+b$ i.e., of $x(x-p)$ subject to $x\in[0,Q]$. The point of (global) minima is given by $x=\left\{\begin{array}{cc} 0,&\text{ if }p\le0\\\frac{p}{2},&\text{ if }0\le p\le2Q\\Q,&\text{ if }p>2Q\end{array}\right.$


May be you have mistyped the question, should it be $(x-x_0)^2+y^2$?

Else (in the current form), the answer is quite easy (unless I missed something). Your objective function (OF) becomes $(Q-y^2-x_0)^2+y^2$ So writing $y^2=p$ and assuming $Q-x_0=a\ge\dfrac{1}{2}$, the OF becomes

$(p-a)^2+p$ $=\left[p-(a-\dfrac{1}{2})\right]^2+a^2-(a-\dfrac{1}{2})^2$ So the minimum value is $\quad a^2-(a-\dfrac{1}{2})^2=a-\dfrac{1}{4}$

  • 0
    @Max, I have edited my answer to fix everything.2011-11-10