Consider the following problem: given $g_1 \ldots g_i,h_1 \ldots h_i \in G$, $\forall i$ find $x_i$ such that $g_i^{x_i}=h_i$
For $i=1$ this is the discrete log problem and is assumed to to have some level of security based on the order of $G$ for certain groups. Let's call this security level $\lambda$.
What happens to $\lambda$ as $i$ increases? Best case, it scales linearly with $i$. That is, its twice as hard to solve two DL instances in the same group $G$ as it is to solve one. Worse case, its no harder.
Im particularly curious with what happens when $G$ is a Schnorr group, though composite groups might work too.
At least on a nieve quick glance, neither Pohlig–Hellman nor Baby-step Giant-step seem to benefit much from being run multiple times in the same group. I'm worried, however, that there are overlapping sub problems that if properly leveraged, perhaps using dynamic programing, would lead to a sub-linear scale up. The question is, how effective these techniques are.
I'd imagine someone has examined this before and I just need to find a paper on it, but as of yet I haven't found one.
UPDATE The Pollard Rho algorithm can apparently solve $i$ dl instances in the same group in $O{\sqrt{Ni}}$ where $N$ is the order of the group. So $\frac{1}{2}\lambda\sqrt{i}$.[1]
The real question appears to be how index calculus methods handle it.