Suppose $n_1$ and $n_2$ are positive integer numbers. How can I approach this problem:
$$\min n_1+n_2 \\~\text{s.t.}~\frac{9}{n_1}+\frac{2}{n_2} \le \epsilon$$
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get some graph paper and carefully draw, say, $$ \frac{9}{x} + \frac{2}{y} = 11 $$ for positive $x,y.$ – Will Jagy Feb 19 '15 at 02:35
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In positive reals, you would have got the min as $\dfrac{(3+\sqrt2)^2}\epsilon$, when $(n_1, n_2) = \dfrac{3+\sqrt2}\epsilon \cdot (3, \sqrt2)$, so you might want to search for solutions nearby... – Macavity Feb 20 '15 at 05:04