> what actually does convex mean in the context of optimization
It means there's a deterministic solution.
Longer explanation:
When you're optimizing a function in math, the function's solutions exist in N-dimensional space. If the space of solutions is a hyperplane in this space, and you want to find a value on that plane that minimizes the function, a convex plane will allow you to find the minimum every time, deterministically -- one way is to simply follow the negative gradient of the plane. A non-convex plane may have lots of holes or other shapes (e.g "saddle points"), so if you follow the negative gradient, you'll end up thinking you're at a minimum, but turns out you're only at a _local_ minimum and there may be another, better solution out there.
So whenever you see "convex" read: a best solution exists.
Whenever you see "non-convex" read: an approximation is your best hope. Motion planning is non-convex.
From https://en.wikipedia.org/wiki/Convex_function: "Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.[4]"
It means there's a deterministic solution.
Longer explanation:
When you're optimizing a function in math, the function's solutions exist in N-dimensional space. If the space of solutions is a hyperplane in this space, and you want to find a value on that plane that minimizes the function, a convex plane will allow you to find the minimum every time, deterministically -- one way is to simply follow the negative gradient of the plane. A non-convex plane may have lots of holes or other shapes (e.g "saddle points"), so if you follow the negative gradient, you'll end up thinking you're at a minimum, but turns out you're only at a _local_ minimum and there may be another, better solution out there.
So whenever you see "convex" read: a best solution exists. Whenever you see "non-convex" read: an approximation is your best hope. Motion planning is non-convex.
From https://en.wikipedia.org/wiki/Convex_function: "Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.[4]"