An essentially decentralized interior point method for control

Decentralized non-convex optimization via bi-level SQP and ADMM

“…(c) The ADMM convergence properties invoked in the proof of Lemma 4 ensure that ADMM can satisfy the modified stopping criterion (10). By Lemma 2, satisfaction of the modified stopping criterion together with the correct active set implies satisfaction of the inexact Newton stopping criterion (9).…”

“…Observe that the difference between F and F is that F does not include the block rows min(−h i (x k i ), µ k i ), i ∈ S to avoid potential differentiability issues outside B ε 1 . We note that the subsystems can evaluate (10) locally and only have to communicate convergence flags, if i E i (x k i + s k i ) = c and if a suitable norm such as • ∞ is chosen. Lemma 2 (Modified stopping criterion).…”

“…In order to apply the modified stopping criterion (10), we show in this subsection that ADMM terminates at the correct active set in a neighborhood of p ⋆ . To this end, we first show that the optimal active sets of (1), (7), and (8) are equivalent in a neighborhood of p ⋆ , if ADMM is initialized appropriately.…”

“…Remark 3 (Local vs. global convergence). As shown in Lemma 2, we may replace the standard inexact Newton stopping criterion by (10), if the current iterates are at the optimal active set of p ⋆ . This is done, because ∇ F can be computed even outside B ε .…”

“…They establish global convergence, but require a feasible initialization which may be difficult to obtain by distributed computation. As a further bi-level algorithm, [10] proposes an essentially decentralized interior point method with local convergence guarantees for general non-convex problems, whereby only scalars have to be communicated among non-neighboring subsystems.…”