I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq x$ and $\mathbf{1}^\top \mathbf{x} = {1}$. This will have to be performed multiple times.

What is a good optimization method to use? Preferably fairly simple to describe and implement?

There are some really fancy methods (such as “Mirror descent and nonlinear projected subgradient methods for convex optimization”, Beck and Teboulle) that specifically have methods for minimizing over the unit simplex. But these methods only use the gradient and not the Hessian.