hafusxl

2022-05-01

Suppose I have $n$ integers ${\mu}_{i}$, $i=\{1,2,...,n\}$. Define $\overline{\mu}=\frac{1}{n}\sum _{i=1}^{n}{\mu}_{i}$. It is given that all the ${\mu}_{i}$'s are either +1 or −1. How can I show that $\sum _{i=1}^{n}({\mu}_{i}-\overline{\mu}{)}^{2}$ is maximized only when $\frac{\lfloor n\rfloor}{2}$ or $\frac{\lceil n\rceil}{2}$ are +1 and rest are −1's.

I was thinking about proving this inductively, but I am not sure how.

I was thinking about proving this inductively, but I am not sure how.

Penelope Carson

Beginner2022-05-02Added 16 answers

Let $\varphi (\mu )=\sum _{k}({\mu}_{k}-\overline{\mu}{)}^{2}=\sum _{k}{\mu}_{k}^{2}-(\sum _{k}{\mu}_{k}{)}^{2}$.

Since $|{\mu}_{k}|=1$, we see that $\varphi (\mu )=n-(\sum _{k}{\mu}_{k}{)}^{2}$.

Hence $\varphi $ will be maximised when $\mu \mapsto (\sum _{k}{\mu}_{k}{)}^{2}$ is minimised.

As an aside, note that this is also the solution to the relaxation $max\{\varphi (\mu )\mid {\mu}_{k}\in [-1,1]\}$, which is convex. It is straightforward to see that $\frac{\mathrm{\partial}\varphi (\mu )}{\mathrm{\partial}{\mu}_{k}}=2({\mu}_{k}-\overline{\mu})$ and $\frac{{\mathrm{\partial}}^{2}\varphi (\mu )}{\mathrm{\partial}{\mu}_{k}^{2}}=2(1-\frac{1}{n})$. In particular, we see that if ${\mu}_{k}\in (-1,1)$ then there is some $t$ such that $\mu +t{e}_{k}$ is feasible and $\varphi (\mu +t{e}_{k})>\varphi (\mu )$. Hence at a maximum, we must have $|{\mu}_{k}|=1$.

Since $|{\mu}_{k}|=1$, we see that $\varphi (\mu )=n-(\sum _{k}{\mu}_{k}{)}^{2}$.

Hence $\varphi $ will be maximised when $\mu \mapsto (\sum _{k}{\mu}_{k}{)}^{2}$ is minimised.

As an aside, note that this is also the solution to the relaxation $max\{\varphi (\mu )\mid {\mu}_{k}\in [-1,1]\}$, which is convex. It is straightforward to see that $\frac{\mathrm{\partial}\varphi (\mu )}{\mathrm{\partial}{\mu}_{k}}=2({\mu}_{k}-\overline{\mu})$ and $\frac{{\mathrm{\partial}}^{2}\varphi (\mu )}{\mathrm{\partial}{\mu}_{k}^{2}}=2(1-\frac{1}{n})$. In particular, we see that if ${\mu}_{k}\in (-1,1)$ then there is some $t$ such that $\mu +t{e}_{k}$ is feasible and $\varphi (\mu +t{e}_{k})>\varphi (\mu )$. Hence at a maximum, we must have $|{\mu}_{k}|=1$.

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