How can I find the maximum value of $x_6-x_1$ subject to the two constraints $\sum_{j=1}^{6} x_j^2$ and $\sum_{j=1}^{6} x_j = 0$

Question

I currently have six variables $x_1, x_2, x_3, x_4, x_5, x_6$. I am trying to determine how large I can make the difference $x_6-x_1$ while satisfying the constraints:

$\sum_{j=1}^{6} x_j^2 \leq 1$

$\sum_{j=1}^{6} x_j = 0$

So far I have tried to use an optimization package in R but to no avail. Is there a way I can program this out or would it be too much to handle? Thanks!

What's the constraint? That those two sums are equal to specific number? 0? (If it's 0, it seems pretty meaningless, since then all the squares would have to be 0)... — Alan, Oct 13 '14 at 00:49
You can solve this by inspection, what do you need an optimization package for? — Kirill, Oct 13 '14 at 00:55

score 3 · Accepted Answer · answered Oct 13 '14 at 00:55

3

Note that $$ (x_6-x_1)^2\leq 2(x_6^2+x_1^2)\le2 $$ with all inequalities becoming equalities when $x_6=\pm1/\sqrt{2}$ and $x_1=-x_6$. Thus, $$ \max(x_6-x_1)=\sqrt{2} $$ which realizes when $x_1=-1/\sqrt{2}$, $x_2=x_3=x_4=x_5=0$, and $x_6=1/\sqrt{2}$.

answered Oct 13 '14 at 00:55

Kim Jong Un

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How can I find the maximum value of $x_6-x_1$ subject to the two constraints $\sum_{j=1}^{6} x_j^2$ and $\sum_{j=1}^{6} x_j = 0$

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