Duality and Optimization for Generalized Multi-Hop MIMO Amplify-and-Forward Relay Networks With Linear Constraints

We consider a generalized multi-hop MIMO amplify-and-forward (AF) relay network with multiple sources/destinations and arbitrarily number of relays. We establish two dualities and the corresponding dual transformations between such a network and its dual, respectively, under single-network linear constraint and per-hop linear constraint. A unified optimization framework is proposed to find a stationary point for an important class of nonconvex optimization problems of AF relay networks based on a local Lagrange dual method, where the primal algorithm only finds a stationary point for the inner loop problem of maximizing the Lagrangian w.r.t. the primal variables. The input covariance matrices are shown to satisfy a polite water-filling structure at a stationary point of the inner loop problem. The duality and polite water-filling are exploited to design fast primal algorithms. Compared with the existing algorithms, the proposed optimization framework with duality-based primal algorithms can be used to solve more general problems with lower computation cost.