Gustavo Lacerda

Let A be a linear space.
Let A_m be the subset of A on which a given non-linear constraint is satisfied (drawn above as a curved 2D surface).

Goal: maximize an objective D within the manifold A_m.

Ideally, we would have a coordinate-system for addressing points in A_m, and optimize as if it were a Euclidean space. But I don't know how to do that.

Still, I can see how I could explore A_m by exploring tangent directions, evaluating D, and optimizing in a fashion similar to how it's done in the standard setting (e.g. to do gradient descent: for any point in A_m, compute the gradient in A, project it onto the tangent plane, and this gives the gradient in A_m... when following this gradient, we will leave A_m slightly, and we'll need to snap back to A_m somehow). I suspect there's room for cleverness in these details.