The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ where $k_i$ are real known constants such that (2) $k_1\ge k_2\ge\cdots\ge k_n=0$ and the unknown real variables satisfy $x_1\ge x_2\ge\cdots\ge x_n>0$. Also (3) $1\le k_1\le 2(n-1)$ and (4) $1\le x_1\le 2(n-1)$. ( Note that conditions (3) and (4) are for the largest of $x_i$ and $k_i$ values). Can anybody with the help of quadratic programming or by any other method(s) suggest an upper bound for the function $f(x_1,x_2,\cdots,x_n)$? I guess that it is $2n(n-2)$. Is it true?
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$\begingroup$ My guess for the upper bound came by taking the example $x_1=2(n-1)$, $,x_2=x_3=\cdots=x_n=n-2$ and $k_1=k_2=k_3=\cdots=k_{n-1}=n$ and $k_n=0$. So that sum of x values and i values each equal to $n(n-1)$ $\endgroup$– shahulhameedCommented Jan 24, 2023 at 17:39
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