Quadratic programming problems, which are a type of convex optimization, typically only have one decision variable $x$
$$\arg \min_x \enspace f(x)$$
Is it also possible to solve for a second decision variable simultaneously?
$$\arg \min_{x,y} \enspace f(x,y)$$
If so, what are some examples and models that do multivariate convex optimization? What are some issues to be aware of when introducing a second decision variable $y$, compared to univariate quadratic programming,?