On the Exponential Number of Connected Components for the Feasible Set of Optimal Decentralized Control Problems

The optimal decentralized control (ODC) problem is known to be NP-hard and many sufficient tractability conditions have been derived in the literature for its convex reformulations or approximations. To better understand the root cause of the non-existence of efficient methods for solving ODC, we propose a measure of problem complexity in terms of connectivity, and show that there is no polynomial upper bound on the number of connected components for the set of static stabilizing decentralized controllers. Specifically, we present a subclass of problems for which the number of connected components is exponential in the order of the system and, in particular, any point in each of these components is the unique solution of the ODC problem for some quadratic objective functional. The results of this paper have two implications. First, the recent effort in machine learning advocating the use of local search algorithms for non-convex problems, which has also been successful for the optimal centralized control problem, fails to work for ODC since it needs an exponential number of initializations. Second, no reformulation of the problem through a smooth change of variables can reduce the complexity since it maintains the number of connected components. On the positive side, we show that structural assumptions can reduce the connectivity complexity of ODC, one such structure is the system being highly damped.