Is it possible to combine projections for constrained optimization?

[vivien000]

Hi! I'd like to solve a constrained optimization problem like:
$$\min f(x)$$
with:

• $x \in \mathcal{R}^{n+}$
• $Ax = b$

I understand it's possible to enforce the first constraint with jaxopt.projection.projection_non_negative and the second one with jaxopt.projection.projection_affine_set. But is it possible to enforce both at the same time? If so, how?

Many thanks in advance and thanks for the great library!

[vivien000]

After having thought about it, if I get it right:

• for the example I gave above, we can just use jaxopt.projection.projection_polyhedron
• in general, we can enforce as many constraints as we like as long as we're able to write the corresponding projection such as in projection.py

[mblondel]

If f is a quadratic function, you can use one of our quadratic programming solvers. If f is a more general function, projected gradient + projection_polyhedron would be indeed one way to do it, although the projection needed on each iteration could be expensive to compute. We don't support it in the library yet but ADMM comes to mind for solving this type of problem.