Log-convexity preserved by sum?

Question

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,

A simpler way to prove this is to use: $\sqrt{ab}+\sqrt{cd} \le \sqrt{a+c}\sqrt{b+d}$ where $a=f(x)$, $b=f(y)$, etc. (hint: $f( (x+y)/2) \le \sqrt{f(x)}\sqrt{f(y)}$) — suvrit, Apr 15 '15 at 15:14
Related: How to prove that the sum of two log-convex functions is log-convex? — Martin R, Sep 13 '20 at 17:54

score 3 · Accepted Answer · answered Apr 15 '15 at 15:27

3

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.

answered Apr 15 '15 at 15:27

GH from MO

1,111

Increasing of function is not necessary. – kaka Dec 10 '20 at 07:44

Log-convexity preserved by sum?

1 Answers1