  We describe a convex programming approach to the calculation of lower bounds
on the minimum cost of constrained decentralized control problems with
nonclassical information structures. The class of problems we consider entail
the decentralized output feedback control of a linear time-varying system over
a finite horizon, subject to polyhedral constraints on the state and input
trajectories, and sparsity constraints on the controller's information
structure. As the determination of optimal control policies for such systems is
known to be computationally intractable in general, considerable effort has
been made in the literature to identify efficiently computable, albeit
suboptimal, feasible control policies. The construction of computationally
tractable bounds on their suboptimality is the primary motivation for the
techniques developed in this note. Specifically, given a decentralized control
problem with nonclassical information, we characterize an expansion of the
given information structure, which ensures its partial nestedness, while
maximizing the optimal value of the resulting decentralized control problem
under the expanded information structure. The resulting decentralized control
problem is cast as an infinite-dimensional convex program, which is further
relaxed via a partial dualization and restriction to affine dual control
policies. The resulting problem is a finite-dimensional conic program whose
optimal value is a provable lower bound on the minimum cost of the original
constrained decentralized control problem.
