When I read somewhere about the importance of convexity for optimization most of the time it deals with the nice property of convex functions that local and global minima are the same.
This is a very neat property to have but if this is the only reason we value convexity so highly then why don't we look more into pseudoconvexity since it is less strict than convexity?
For the definition of pseudoconvexity, I will cite Wikipedia:
Consider a differentiable function$ f:X\subseteq \mathbb {R} ^{n}\rightarrow \mathbb {R}$, defined on a (nonempty) convex open set$X$ of the finite-dimensional Euclidean space$\mathbb {R} ^{n}$. This function is said to be pseudoconvex if the following property holds:
$$ \forall x,y\in X:\quad \nabla f(x)\cdot (y-x)\geq 0\Rightarrow f(y)\geq f(x).$$
Pseudoconvex functions also have the property that local and global minima are the same. Otherwise we have:$$Convex \Rightarrow pseudoconvex \Rightarrow quasiconvex$$ There has to be a reason that pseudoconvexity isn't discussed that much.