TITLE: Why is convexity more important than quasi-convexity in optimization?
QUESTION [20 upvotes]: In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have globally optimal solutions, so one can use methods such as gradient (steepest) descent to find such solutions, but I am not convinced.
For example, the function $|x|^{0.001}$  is not convex (see the yellow-shaded area in the picture below) but it has a single global minimizer.
                      
So why is convexity so important in optimization? Why is e.g. quasiconvexity often not enough?

REPLY [2 votes]: Like someone else said, convex problems often have a dual, which lets an algorithm solve the problem from two sides and declare the solution when the duality gap is sufficiently close to 0 (when the lower and upper bound on the solution converge). There are also fairly efficient methods and algorithms that can solve canonical convex problems which do not extend easily to general quasiconvex problems.
For strict quasiconvex problems, a gradient descent method should be sufficient, but it's a question of algorithmic efficiency. The algorithms for canonical convex problems are just much more efficient.
At least, that's my impression of why there's so much focus on convexity. However, one could certainly argue that quasiconvexity is often overlooked when it would be useful...