Is constrained least squares regression convex?

Question

Would anyone know if the following optimisation can be a 'convex' optimisation with a global minimum under certain conditions if any?

minimise w.r.t x, || Ax - b ||^2

such that

f(x) <=0

g(x) = 0

?

For example would the above be convex with a global minimum if A was a positive definite matrix?

Any pointers would be much appreciated.

Thanks.

Answer 1

That would depends on the property of $f$ and $g$. If $f$ is convex and $g$ are affine, then the domain is convex.

$\|Ax-b\|^2$ is always convex regardless of whether $A$ is positive definite.