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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.

enter image description here

  • The trivial solution is to solve $\min_{x,y} f(x)+g(y)$ and use $(x+y)/2$ as the result. This can be generalized to more functions $f_1,\dots,f_k$. –  Jun 15 '19 at 06:11
  • @Rahul Solving what you have proposed, implicitly means that solve them separately and find the mean of all solutions? right? –  Jun 15 '19 at 06:27
  • Yes, that's why I called it the trivial solution. –  Jun 15 '19 at 07:03
  • I think @Rahul’s approach, though trivial, is also optimal, at least as the problem is currently stated. – LarrySnyder610 Jun 15 '19 at 14:10
  • I don't believe it is likely you can do better than the trivial solution. Of course, from a practical point of view you could be solving the separate optimization problems in parallel and gathering the results to the final minimizer. – Michael Grant Jun 15 '19 at 23:10
  • @Michael Grant: For trivial solution I have to solve $n$ different optimizations. However, I need a way to define one optimization problem including all $f_i$'s that gives me the optimal point which is the one that Rahul pointed out. That would be interesting. –  Jun 17 '19 at 03:34
  • Yes but you can always combine separate problems trivially like Rahul did. I'm saying I doubt you can do better. – Michael Grant Jun 17 '19 at 03:38

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