Convergent Proximal Multiblock ADMM for Nonconvex Dynamics-Constrained Optimization

This paper proposes a provably convergent multiblock ADMM for nonconvex optimization with nonlinear dynamics constraints, overcoming the divergence issue in classical extensions. We consider a class of optimization problems that arise from discretization of dynamics-constrained variational problems that are optimization problems for a functional constrained by time-dependent ODEs or PDEs. This is a family of $n$-sum nonconvex optimization problems with nonlinear constraints. We study the convergence properties of the proximal alternating direction method of multipliers (proximal ADMM) applied to those problems. Taking the advantage of the special problem structure, we show that under local Lipschitz and local $L$-smooth conditions, the sequence generated by the proximal ADMM is bounded and all accumulation points are KKT points. Based on our analysis, we also design a procedure to determine the penalty parameters $ρ_i$ and the proximal parameters $η_i$. We further prove that among all the subsequences that converge, the fast one converges at the rate of $o(1/k)$. The numerical experiments are performed on 4D variational data assimilation problems and as the solver of implicit schemes for stiff problems. The proposed proximal ADMM has more stable performance than gradient-based methods. We discuss the implementation to solve the subproblems, a new way to solve the implicit schemes, and the advantages of the proposed algorithm.