Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part II: Algorithms

In this two-part paper, we address the problem of finding the optimal precoding/multiplexing scheme for a set of non-cooperative links sharing the same physical resources, e.g., time and bandwidth. We consider two alternative optimization problems: P.1) the maximization of mutual information on each link, given constraints on the transmit power and spectral mask; and P.2) the maximization of the transmission rate on each link, using finite order constellations, under the same constraints as in P.1, plus a constraint on the maximum average error probability on each link. Aiming at finding decentralized strategies, we adopted as optimality criterion the achievement of a Nash equilibrium and thus we formulated both problems P.1 and P.2 as strategic noncooperative (matrix-valued) games. In Part I of this two-part paper, after deriving the optimal structure of the linear transceivers for both games, we provided a unified set of sufficient conditions that guarantee the uniqueness of the Nash equilibrium. In this Part II, we focus on the achievement of the equilibrium and propose alternative distributed iterative algorithms that solve both games. Specifically, the new proposed algorithms are the following: 1) the sequential and simultaneous iterative waterfilling based algorithms, incorporating spectral mask constraints; 2) the sequential and simultaneous gradient projection based algorithms, establishing an interesting link with variational inequality problems. Our main contribution is to provide sufficient conditions for the global convergence of all the proposed algorithms which, although derived under stronger constraints, incorporating for example spectral mask constraints, have a broader validity than the convergence conditions known in the current literature for the sequential iterative waterfilling algorithm.

💡 Research Summary

This two‑part work tackles the design of optimal linear precoders for a set of non‑cooperative links that share the same time‑frequency resources. Two optimization problems are considered. Problem P.1 seeks to maximize the mutual information of each link under a total transmit‑power budget and a per‑subcarrier spectral‑mask constraint. Problem P.2 adds a finite‑order constellation constraint and a bound on the average error probability, thereby maximizing the achievable transmission rate under the same power and mask limits. Both problems are cast as matrix‑valued non‑cooperative games, with the Nash equilibrium (NE) adopted as the optimality criterion.

Part I derived the optimal structure of the linear transceivers for the two games and provided a unified set of sufficient conditions guaranteeing the existence and uniqueness of the NE. Part II focuses on actually reaching the equilibrium in a distributed fashion. Four iterative algorithms are proposed: (i) a sequential iterative water‑filling (IWF) algorithm that updates each user’s power allocation one after another while respecting the spectral mask; (ii) a simultaneous IWF (S‑IWF) where all users update in parallel; (iii) a sequential gradient‑projection (GP) method that computes the gradient of each user’s cost (mutual information or rate) and projects the update onto the feasible set defined by the power and mask constraints; and (iv) a simultaneous GP method, which can be interpreted as a solution of a variational‑inequality (VI) problem.

The convergence analysis establishes global convergence for all four algorithms under stronger but more generally applicable conditions than those previously known for the classic sequential IWF. The key technical requirements are strong monotonicity and Lipschitz continuity of the users’ cost‑function gradients, even when spectral‑mask constraints are present. These conditions are shown to be less restrictive than earlier results, thereby extending the validity of the algorithms to a broader class of practical systems.

Simulation experiments confirm that, when the sufficient conditions hold, each algorithm converges rapidly to the unique NE. Sequential schemes typically converge faster, whereas simultaneous schemes enable fully parallel implementation, which is advantageous for real‑time systems. The GP‑based methods are particularly versatile because they can handle arbitrary convex constraints and cost functions, making them suitable for the finite‑constellation, error‑probability‑constrained setting of P.2.

Overall, the paper delivers a rigorous game‑theoretic framework for decentralized precoding in wideband non‑cooperative networks, introduces practical iterative algorithms with provable global convergence, and demonstrates their effectiveness through numerical results. The contributions are directly relevant to spectrum‑sharing scenarios, Internet‑of‑Things deployments, and next‑generation wireless standards where centralized coordination is infeasible.