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I have the following constraint $$ x^TAx+b^Tx=c $$ where $A$ is a positive definite matrix. Is there any way to take advantage of the strict convexity of this expression to reformulate the constraint as something simpler?

Thoughts/attempt:

I was thinking we could form $$B = A^{1/2}$$ and maybe introduce a change-of-variable as a start, but I'm not sure if this leads anywhere.

jonem
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  • This is not a "strictly convex equality constraint." There is no such thing. It is an equality constraint involving a strictly convex function, yes. But the set it describes is non-convex, and that cannot be fixed by rewriting. – Michael Grant Jun 26 '15 at 12:25
  • You may find it illuminating to rewrite the expression in terms of $y = A^{1/2} x$ with a symmetric positive definite root of $A$, along the lines you were thinking. This will show you where that line of thought leads. – hardmath Jun 26 '15 at 13:14

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