Maximize the sum of numbers subject to an upper bound on their sum of squares

Question

I want to find a maximum for $\sum_i^n x_i$ subject to $\sum_i^n x_i^2 \leq c$, where c is some constant.

I've done some searching and that's hinted at using the Cauchy-Schwarz inequality, but I don't yet see how to use it here.

Thanks for the hints guys; I smacked myself on the forehead when I saw these! — Gaussian0617, Oct 20 '15 at 14:59

Chappers · Accepted Answer · 2015-10-20T16:05:04.450

1

$$ \left\lvert \sum_i 1x_i \right\rvert^2 \leqslant \left( \sum_i 1 \right) \left( \sum_i x_i^2 \right) \leqslant nc, $$ applying Cauchy-Schwarz. Equality occurs precisely when the vectors $(1,1,\dotsc,1)$ and $(x_1,x_2,\dotsc,x_n)$ are parallel. It's easy after that.

edited Oct 20 '15 at 16:05

answered Oct 20 '15 at 14:25

Chappers

67,606

Thanks Chappers. Just one minor clarification; shouldn't the equality on the right be a $\leqslant$ as well? – Gaussian0617 Oct 20 '15 at 15:01
Yeah, probably... – Chappers Oct 20 '15 at 16:04

Maximize the sum of numbers subject to an upper bound on their sum of squares

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