I am interested in minimizing $$min_{x \in R^{n^l}} f^l(x),$$ where $f^l(x)$ is nonlinear objective function arising from discretization of PDE. I would like to use nonlinear multilevel minimization techniques, such as MG-opt (Nash, A multigrid approach to discretized optimization problems, https://www.tandfonline.com/doi/full/10.1080/10556780008805795?scroll=top&needAccess=true ).
MG-opt is generalization of full approximation scheme to minimization problems. Hence, on coarse level, it minimizes $$min_{x \in R^{n^L}} h^L(x) = f^L(x) + \langle R \nabla f^l(x^l_k) - \nabla f^L(R x^l_k) \rangle$$ in order to obtain coarse grid correction. This objective function is built as a sum of rediscretized fine level physics ($f^L$) and first order consistency term (difference between restricted fine level gradient and initial coarse level gradient).
What is correct way to treat boundary conditions, such that minimization on coarse level is feasible (respects coarse level BC), but does not hinders the convergence properties of the method. Should coarse level objective function be computed over whole domain, including boundary conditions of the original physics?
