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Can one have a condition like this in semidefinite programming?
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thiscanconditionsemidefiniteprogramminglikeonehave
Problem
Is it possible to have the following condition in a semidefinite programming as a constraint?
$
M=
\left[ {\begin{array}{cc}
a & \sqrt{u} \\
\sqrt{u} & b \\
\end{array} } \right]
\geq 0$
where $\geq 0$ means positive semidefinite.
$
M=
\left[ {\begin{array}{cc}
a & \sqrt{u} \\
\sqrt{u} & b \\
\end{array} } \right]
\geq 0$
where $\geq 0$ means positive semidefinite.
Solution
Assuming that a, b, and u are real variables in your semidefinite program, the answer is negative.
The important fact here is that each constraint in a semidefinite program defines a convex set. Your condition cannot be written as a constraint in a semidefinite program because the set of points (a, b, u) ∈ ℝ3 that satisfy the condition is not convex. I suggest that you try to prove that this set is indeed not convex, but here is a proof (put the mouse cursor in the box below to show the proof):
Proof: (a, b, u) = (1, 1, 1) and (a, b, u) = (3, 3, 9) satisfy the condition, but their midpoint (a, b, u) = (2, 2, 5) does not satisfy the condition.
The important fact here is that each constraint in a semidefinite program defines a convex set. Your condition cannot be written as a constraint in a semidefinite program because the set of points (a, b, u) ∈ ℝ3 that satisfy the condition is not convex. I suggest that you try to prove that this set is indeed not convex, but here is a proof (put the mouse cursor in the box below to show the proof):
Proof: (a, b, u) = (1, 1, 1) and (a, b, u) = (3, 3, 9) satisfy the condition, but their midpoint (a, b, u) = (2, 2, 5) does not satisfy the condition.
Context
StackExchange Computer Science Q#9724, answer score: 5
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