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I'm currently reading Boyd's book on Convex Optimization (free book which can be found here : http://stanford.edu/~boyd/cvxbook/) , and there is one proof I do not understand : p. 105, the book says that the sum of two log-convex functions is log-convex

Let $f$ and $g$ be log-convex, then $F=\log f$ and $G=\log g$ are convex, from the composition rules for convex functions, it follows that $\log (\exp F+\exp G)$ is convex.

I don't see how he applies the composition rules, since in all versions of the rule (p. 83-86), the composition $h \circ g$ is convex if $h$ is convex, but here, $\log$ is concave. What am I missing ?

Thanks in advance,

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The author meant that the function $(u,v)\mapsto\log(\exp u+\exp v)$ is convex and increasing, hence composing it with the convex function $x\mapsto (F(x),G(x))$ yields the convex function $x\mapsto\log(\exp F(x)+\exp G(x))$. This is a special case of Example 3.14 on Page 87.

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  • $\begingroup$ Increasing of function is not necessary. $\endgroup$
    – kaka
    Dec 10, 2020 at 7:44

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