I'm looking for an optimization algorithm that solves a problem with a multiplied term. For example,
$$ \min_{x \in \mathbb{R}, z \in \mathbb{R}^{3}} \lVert xz + az + b\rVert_{2}^{2} $$ for given $a \in \mathbb{R}$ and $b \in \mathbb{R}^{3}$ constant.
I found a related question and answer in which ADMM is recommended.
So I've been scheming ADMM but it seems like it only works for a problem with "separable term" like $\min_{x, z} f(x) + g(z) \text{ subject to ...}$. And I don't think that my problem is not in the case.
Does it make sense as reported in the answer? If so, is there any theory for such problems (e.g. convergence analysis)?
Choose $z = b$ and choose $x$ so that $x + a = -1$. Then the objective function value is $0$, so the optimization problem has been solved.