Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems (1302.0756v2)

Abstract: We propose a novel decomposition framework for the distributed optimization of general nonconvex sum-utility functions arising naturally in the system design of wireless multiuser interfering systems. Our main contributions are: i) the development of the first class of (inexact) Jacobi best-response algorithms with provable convergence, where all the users simultaneously and iteratively solve a suitably convexified version of the original sum-utility optimization problem; ii) the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based, instead, on heuristics; and iii) a framework that can be easily particularized to well-known applications, giving rise to very efficient practical (Jacobi or Gauss-Seidel) algorithms that outperform existing adhoc methods proposed for very specific problems. Interestingly, our framework contains as special cases well-known gradient algorithms for nonconvex sum-utility problems, and many blockcoordinate descent schemes for convex functions.

• The paper introduces inexact Jacobi best-response algorithms with provable convergence for distributed optimization.
• It proposes a dynamic pricing mechanism based on variational inequalities to efficiently manage interference among users.
• The framework adapts to various scenarios, demonstrating faster convergence in SISO and MIMO channels and broad applicability.

Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems

The paper "Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems" by Gesualdo Scutari et al. presents a novel framework for distributed optimization in nonconvex sum-utility problems involving multi-agent systems. This research focuses on addressing the computational complexities associated with distributed optimization, particularly in environments characterized by user interference and nonconvex objective functions.

The primary contributions of the paper are threefold. Firstly, it introduces the first class of inexact Jacobi best-response algorithms with provable convergence. These algorithms allow users to simultaneously and iteratively solve a convexified version of their respective optimization problems. A key advantage of this approach is its capacity to support distributed implementation with local information, a significant step forward in managing the complexity and overhead commonly associated with centralized networks.

Secondly, the authors propose a generalized dynamic pricing mechanism that offers a unifying perspective on existing pricing strategies typically grounded in heuristic approaches. Rather than resorting to fixed pricing strategies, the proposed mechanism dynamically alters prices based on user interactions, leading to more efficient resource allocation and interference management. The pricing mechanism is grounded in variational inequalities and game theory, enabling users to optimize their individual utility while accounting for the effects of their actions on others in the network.

Thirdly, the proposed framework allows easy adaptation to specific problem instances, exhibiting notable efficiency in several well-known applications. This versatility derives from the framework's ability to include, as special cases, existing gradient algorithms for nonconvex sum-utility problems and many block-coordinate descent schemes for convex functions. This indicates its broad applicability and potential for enhancing existing optimization techniques.

One striking detail of the research is the relaxation of the Successive Convex Approximation (SCA) paradigm. Traditional SCA methods require a tight global upper bound of the objective function, which can be challenging to establish in nonconvex settings. The proposed decomposition approach, however, sidesteps this requirement by employing partial linearizations that exploit any existing convex structures within the problem, leading to more flexible and effective optimization solutions.

The paper empirically demonstrates the efficacy of the proposed algorithms in applications such as resource allocation in both Single-Input Single-Output (SISO) and Multiple-Input Multiple-Output (MIMO) interference channels. Compared to state-of-the-art methods, the proposed approach exhibits faster convergence and improved practical performance without additional computational cost.

The implications of this research are profound both in theory and practice. Theoretically, it expands the domain of distributed optimization by providing tools to solve broader classes of nonconvex problems efficiently. Practically, it offers a scalable solution to multi-agent system optimization, a critical need in modern wireless networks and cognitive radio systems.

Looking forward, this work opens avenues for further research in distributed optimization. Specifically, it might encourage investigations into adaptive mechanisms for parameter selection and further extensions of the framework to scenarios involving long-term channel statistics or stochastic environments. Moreover, the insights from this paper could be instrumental in developing optimization strategies for emerging network architectures, such as 5G and beyond, where distributed and efficient resource management is paramount.

Overall, this paper lays a solid foundation for future advancements in the field of distributed algorithm design, particularly in environments marred by interference and nonconvexity.