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How do we seek certain properties for a solution of a convex minimization problem. For example we want to make sure if the below objective has a symmetric optimal solution:

\begin{equation} \min_X f(X) \text{ s.t. } X1=1, X\ge 0 \end{equation}

for some convex function $f$.

My question is about the general steps one should take for such a goal.

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