Suppose I would like to solve an unconstrained optimization problem $$ \min_{x \in \mathbb{R}^n} \space f(x), $$ using the ellipsoid method where $n$ is small, i.e. $n = 2$, and $f$ is a strictly convex function having a minimizer. Is there a good algorithm to find the initial ellipsoid, namely, an ellipsoid which contains the minimizer? We may possibly impose additional assumptions on $f$ (e.g. strong convexity, Lipschitz continuity, something other growth or rate of change condition), but without assuming it is twice differentiable.
As an example, when $n=1$ and $f$ is continuously differentiable, ellipsoids are intervals, and we can find an initial interval by finding the smallest $k$ for which $f'(2^k) > 0$ and $f'(-2^k) < 0$, meaning that interval $[-2^k, 2^k]$ must contain our minimizer.
