I'd like to find the minimum of a smooth, continuous function inside the unit hypercube (the dimensionality of which could go into the hundreds or even thousands). The function is convex (Hessian $\geq 0$ everywhere) within the unit hypercube but not outside of it. Additionally, I know that the minimum of the function is always positive.
I can evaluate both the function and its gradient relatively cheaply; second derivatives are quite a bit more expensive.
Frank-Wolfe works, and is easy to do as I can always go in the direction of the vertex closest to the negative gradient, but the line search part is really quite slow. It also doesn't seem to make particularly good use of the convex structure or the bound on the minimum.
Are there any other approaches I should research given my problem?