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I am given the following set

$$\Omega_{1} = \{ (x,y) \in \mathbf{R}^{n} \times \mathbf{R}^{n} | \Vert y \Vert_{2}^{2} \leq 10 + x^{T}y - \Vert x \Vert^{2}_{2} \}$$

and would like to determine whether it is convex. I know that the set

$$\Omega = \{ x \in \mathbf{R}^{n} | \Vert x \Vert_{2}^{2} \leq 10 \}$$

is convex. Can I use this fact to simplify the problem since the intersection of half spaces is a convex set? Otherwise, do I need to consider for example

$$ z = \lambda t + (1- \lambda)s$$ $$ w = \lambda p + (1-\lambda)q$$

where $t,s,p,q \in \Omega$ and show that $z, w$ are also in $\Omega_{1}$?

Rodrigo de Azevedo
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Bee
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1 Answers1

4

$\Vert y \Vert_{2}^{2} \leq 10 + x^{T}y - \Vert x \Vert^{2}_{2}$

Is the same as

$\Vert y-x/2\Vert^2+\frac{3}{4}\Vert x\Vert^2\leq 10$

Change variable $u=y-x/2$, $u=\sqrt{\frac{3}{4}}x$.

$\Vert z\Vert^2+\Vert v\Vert^2=\Vert(u,v)\Vert^2\leq 10$

If you know how to show that $\Omega$ is convex, then $\Omega_1$ is the same, just in a space of higher dimension and after a linear change of coordinates.

SphericalTriangle
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