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Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries, and consider the convex optimization problem

$\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$ such that $\sum_{j=1}^k x_j = 1$ and $x_j\ge 0$ for all $j=1...k$.

Assume that $R$ is such that this problem has a unique solution $x$ and that this solution is an interior solution, i.e., $x_j>0$ for all $j=1...k$.

Now assume that the top-left entry of $R$ is raised to some new value $r_{11}'>r_{11}$, giving a new matrix $R'$. Assume that the new solution $x'$ to the above problem is still unique and interior.

Conjecture: $x_1' \ge x_1$.

Task: Prove the conjecture or give a counterexample!

  • By saying that $R$ has positive entries, you mean that $r_{i,j}>0$ for every $i,j$ correct? – Surb Nov 25 '15 at 13:05
  • Is this really a research question? – suvrit Nov 25 '15 at 14:18
  • @Suvrit: Yes, of course it is a research question, why wouldn't it? Do you mean to imply it is too simple? Maybe I don't see an obvious proof? –  Nov 26 '15 at 07:15

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