How to solve this problem?
$ \min_{\mathbf{x},\mathbf{y}}f(\mathbf{x},\mathbf{y})\\ s.t. \mathbf{x}\in\mathcal{X},\mathbf{y}\in\mathcal{Y} $
where $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$, $\mathcal{X}$and $\mathcal{Y}$ are two convex sets, $f(\mathbf{x},\mathbf{y})$ is convex w.r.t. $\mathbf{x}$ but not w.r.t. $\mathbf{y}$.
In general there is no guarantee that you would find the optimal solution for this problem. If f is strongly convex with respect to x then you can find optimal x for each y and the use that x to take the gradient with respect to y ( as the optimizer is unique the function has gradient based on Danskin's theorem: https://en.wikipedia.org/wiki/Danskin%27s_theorem). In order for the gradient to be smooth as well you need the function in x to be strongly convex and the gradients with respect to x and y to be Lipchitiz on x and y (this is easy to prove and you should be able to find a simple proof for it in the books; but in lemma 1 of this paper: https://arxiv.org/abs/1710.10571 that I was coincidentally reading today it proves it for the maximizing strongly concave function). So you can do gradient descent on y with step-size 1 over liptchitz constant and converge to a stationary solution.