Suppose we have a set of strictly convex functions $\{f_1,\cdots,f_k\}$ whose minimizers set is $\{x_1,\cdots,x_k\}$, $x_i$ corresponding to $f_i$.
My main question is finding a way to characterization of a minimization problem whose solution is a point that is closest to all minimizer. But for the time being I skip my question and want to think of the following:
Suppose that we have two convex cost functions $f(x)=x^2$ and $g(x)=(x-1)^2+1$ that have minimizers at $x=0$ and $x=1$ respectively. How can we characterize a minimization problem whose solution is a point that is closest to $x=0$ and $x=1$, i.e., $x=\frac{1}{2}$? To have better understanding, I attached the following figures.
Note: I want to solve one optimization problem containing $f_i$'s converging to the noted point, we do not know what are the minimizers because otherwise we have to solve $n$ different optimizations.
