'Partial Minimizer' of a convex function

Question

Let $f(x,y)$ be a convex function (not necessary differentiable) from $\mathbb R^{m+n} \rightarrow \mathbb R$. Suppose $(x^\star, y^\star)$ satisfies the following conditions: \begin{align*} y^{\star} = \text{argmin}_y f(x^\star, y)\\ x^{\star} = \text{argmin}_x f(x, y^\star).\\ \end{align*}

Can we conclude that $(x^\star, y^\star)$ is a minimizer of $f$?

If not, what would be the answer if we further assume that $f(x,y)$ is also convex on both $x$ and $y$ when fixing the other?

Thanks!

Answer 1

This is not true. Take $f \colon \mathbb R^2 \to \mathbb R$ defined via $$ f(x,y) := |x - y| - \frac12 \, x.$$

This function is convex and $(x^*, y^*) = (0,0)$ satisfies your condition. However, $(0,0)$ is not a minimizer of $f$.