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Given the symmetric positive definite matrix $\bf A$,

$$ \begin{array}{ll} \underset {{\bf x}} {\text{minimize}} & {\bf x}^\top {\bf A} \, {\bf x} \\ \text{subject to} & {\bf 1}^\top {\bf x} = 1 \\ & {\bf x} \geq {\bf 0} \end{array} $$

I am looking for an analytical solution of the (convex) quadratic program above. I am not looking for a QP solver, I am aware of their existence and their performance. I am looking for an analytical solution, or a theorem proving that there is none. Is anybody aware of any solution? Or a starting point?

Rodrigo de Azevedo
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mtiret
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    Did you try KKT? – Rodrigo de Azevedo Nov 16 '19 at 10:03
  • yes, but not successfully. The solution is well known without the inequality constraints and is easy to find... – mtiret Nov 17 '19 at 11:09
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    There is no analytic solution. I do not have a proof—and it seems a bit strange in my mind to even expect one! But I will be happy if someone were to prove me wrong :-) – Michael Grant Nov 20 '19 at 16:27
  • @MichaelGrant thank you for your answer, it is what I was afraid of... Just to clarify, I was looking for an analytical solution because numerical solutions are difficult to generalise and then interpret (in a research point of view): if A varies, I have to find the solution for every A... Well, I will do with that then! Huge fan of CVX by the way ;) – mtiret Nov 21 '19 at 10:08

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