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I'm doing some convex optimisation where I'm minimising sum function $f(x) = \sum g_i(x)$, where the $g$'s are convex (and hence so is $f$) and the sum is finite.

In doing so it turns out that $f$ is a linear function between any two (global) optima. Apparently $f$ being linear here implies that the $g$'s are also linear, why is this so?

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    If f is linear, then f is convex as well as concave.2012-12-27

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