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Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0, 1]$. I want to find the global optimum of

$$\min_{x \in [0,1]} a f_1(x) + b f_2(x)$$

for given $a, b \in \mathbb{R}$. Is there a simple solution to this?

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    It's probably easier to answer if a and b are nonnegative, since $af_1(x)+bf_2(x)$ is convex when $a>0$ and $b>0$.2012-10-05
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    Just to add to Snowball's answer more explicitly, if a or b are negative then convexity is not guaranteed so in general it will be difficult to optimize.2012-10-05
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    Moreover, if $a$ and $b$ are nonnegative and $f_1$ and $f_2$ have their global minima at $s$ and $t$, then $f = a f_1 + b f_2$ has a global minimum somewhere in the interval $[\min(s,t),\max(s,t)]$.2012-10-05
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    To expand on @Bitwise's comment: any $C^2$ function on $[0,1]$ can be written as the difference of two $C^2$ convex functions.2012-10-05
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    @RobertIsrael thanks Robert, I wasn't aware of that. Can you provide a reference?2012-10-06
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    Write $f''$ as the sum between its non-negative part and its non-positive part and integrate twice...2012-10-06

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