# convex optimization – Under what condition can we prove \$nabla_x min_y f(x,y)=nabla_x f(x,y^*)\$ where \$y^*=argmin_y f(x,y)\$?

Let $$f: mathbb R^ntimes mathbb R^mto mathbb R$$ be a function. I wonder under what condition can we prove $$nabla_x min_y f(x,y)=nabla_x f(x,y^*)$$ where $$y^*=argmin_y f(x,y)$$. For example, when $$f(x,y)$$ is jointly convex in $$(x,y)$$, then it is known that the above property holds. However, such assumption might be too strong and unnecessary.

1. If $$f(x,y)$$ is only convex in $$y$$, does the above property hold?
2. If $$f(x,y)$$ is non-convex, does the above property hold?
3. If Item 1 or 2 is false, can we relax the property so that $$nabla_x min_y f(x,y)=0$$ iff $$nabla_x f(x,y^*)=0$$?

It seems that the answer is not clear on the website, so I come here for help. Thank you very much!