Minimax Duality for MIMO Interference Networks

A minimax duality for a Gaussian mutual information expression was introduced by Yu. An interesting observation is the relationship between cost constraints on the transmit covariances and noise covariances in the dual problem via Lagrangian multipliers. We introduce a minimax duality for general MIMO interference networks, where noise and transmit covariances are optimized subject to linear conic constraints. We observe a fully symmetric relationship between the solutions of both networks, where the roles of the optimization variables and Lagrangian multipliers are inverted. The least favorable noise covariance itself provides a Lagrangian multiplier for the linear conic constraint on the transmit covariance in the dual network, while the transmit covariance provides a Lagrangian multiplier for the constraint on the interference plus noise covariance in the dual network. The degrees of freedom available for optimization are constituted by linear subspaces, where the orthogonal subspaces induce the constraints in the dual network. For the proof of our duality we make use of the existing polite water-filling network duality and as a by-product we are able to show that maximization problems in MIMO interference networks have a zero-duality gap for a special formulation of the dual function. Our minimax duality unifies and extends several results, including the original minimax duality and other known network dualities. New results and applications are MIMO transmission strategies that manage and handle uncertainty due to unknown inter-cell interference and information theoretic proofs concerning cooperation in networks and optimality of proper signaling.