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.


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.

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  • $\begingroup$ However, if there are some constraints over $a$, $b$ and $u$ it can be, right? $\endgroup$ – Alex Grilo Feb 13 '13 at 19:44
  • $\begingroup$ @Alex Grilo: Clearly it all depends on what kind of constraints you are thinking of. For example, if one of a, b, and u is a constant, then the condition in the question can be easily written as a constraint in a semidefinite program, although this is a silly example. $\endgroup$ – Tsuyoshi Ito Feb 13 '13 at 20:57

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