I am interested in maximizing a function $f: \mathbb{R}^n\rightarrow\mathbb{R}$ over a convex domain $C$. The function $f$ satisfies the following property:
$$ \alpha \cdot \nabla f(x) \leq \nabla f(y),$$
where $\alpha \in [0, 1]$ is a positive constant and $x \leq y$ coordinate-wise. It is easy to see that if $\alpha = 1$ the function is convex. The converse is also true.
Are there results that look at the previous definition and show how well one can optimize the function depending on the constant $\alpha$?
Thanks in advance