Intuitive answer needed for the condition on the sets to be convex

Question

I have read that an optimization problem is convex if the objective function is convex and if all the constraints define convex sets. Is there any intuitive reasoning for sets to be convex? Many thanks in advance.

Answer 1

Convexity allows one to conclude global optimality from local optimality. Local optimality can be concluded using only local information such as the gradient and the Hessian.

In computer science, the equivalent to the class of (continuous) convex optimization problems is arguably the class of (discrete) optimization problems that can be solved using greedy algorithms (which also use only local information). You may want to take a look at greedoids.