Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part II: Algorithms (0707.0871v1)

Abstract: 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.

• The paper introduces iterative waterfilling and gradient projection methods that efficiently achieve Nash equilibria in decentralized wideband systems.
• The study compares sequential (Gauss-Seidel) and simultaneous (Jacobi) schemes, demonstrating faster convergence with the Jacobi approach under spectral radius conditions.
• Numerical simulations validate improved convergence rates and robustness, underscoring practical benefits for distributed wireless network design.

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

The paper under consideration focuses on the development and analysis of decentralized algorithms for optimal linear precoding strategies in wideband non-cooperative systems, contextualized within the framework of game theory. This continuation from the previous part extends the theoretical findings on Nash equilibria applied to multi-user communication systems.

Overview

The research targets two optimization problems: maximization of mutual information and maximization of transmission rate, each constrained by transmit power and spectral mask regulations. The paper utilizes a strategic non-cooperative game theory approach, decomposing multi-user complex interaction into matrix-valued games, further simplified to vector power control problems. These formulations align with Nash equilibria to point out optimal strategies.

Key Contributions

This second part emphasizes algorithmic development for reaching the Nash equilibria in a distributed manner. The primary contributions include:

1. Iterative Waterfilling Algorithms: The authors propose two variants:
   • Sequential Iterative Waterfilling Algorithm (IWFA): This method adheres to a Gauss-Seidel type update scheme where users adjust their strategies sequentially. The convergence analysis stipulates that the spectral radius of a specific matrix must be less than one, ensuring global convergence.
   • Simultaneous Iterative Waterfilling Algorithm: Implementing a Jacobi-based scheme, all users update their strategies simultaneously. This method guarantees faster convergence, which is formally proven to hold under the same conditions as its sequential counterpart.
2. Gradient Projection Algorithms: The research introduces iterative gradient projection methods for rate maximization in a non-cooperative game setting:
   • Simultaneous Iterative Gradient Projection Algorithm: Deploying the Jacobi scheme, it iteratively projects probability vectors onto feasible sets considering the gradient of objective functions.
   • Sequential Iterative Gradient Projection Algorithm: Based on the Gauss-Seidel scheme, it sequentially updates user strategies to achieve the Nash equilibrium.

Numerical Validation and Implications

The algorithms are validated through numerical simulations demonstrating improved convergence rates and robustness compared to existing literature. The simultaneous IWFA, in particular, shows enhanced convergence speed, especially as the number of active users increases.

The implications for practical implementations in wireless networks are noteworthy. These algorithms allow for decentralized operation, reducing the need for a centralized controller and aligning well with distributed architectures seen in modern wireless networks. This approach is practical in dynamically changing environments like cognitive radio networks, optimizing spectrum use without explicit cooperation.

Conclusion

The research presents a compelling suite of distributed algorithms addressing optimal precoding in non-cooperative settings through game-theoretic lenses. By bridging the gap between theoretical derivations of Nash equilibria and practical, efficiently implementable algorithms, this paper significantly progresses toward practical deployment in real-world communication systems. The introduction of memory elements and the variational inequality formulation open avenues for further refinements to manage channel uncertainties and asynchronous updates effectively.

This paper stands to make substantial contributions not just theoretically but also in operational frameworks within contexts demanding decentralized control. Future research directions could explore extending these methods in adaptive environments and convergence proofs in broader signal configurations.