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Robust Beamforming for Downlink Multi-Cell Systems: A Bilevel Optimization Perspective

Xingdi Chen, Yu Xiong, Kai Yang

TL;DR

This work tackles robust downlink beamforming in multi-cell MU-MISO under bounded CSI errors by reformulating the worst-case weighted sum-rate maximization as a bilevel optimization problem. It introduces BLRBF, a cutting-plane based centralized algorithm, and BLADRBF, an asynchronous distributed variant, both with convergence guarantees and no reliance on computationally intensive SDP relaxations. The approach delivers superior robustness and scalability, outperforming SDP-based methods, especially at high SNR or in large-scale networks, while enabling parallel processing across base stations. The results suggest a practical, scalable pathway for robust inter-cell cooperative beamforming in realistic CSI-uncertain environments, with potential extensions to MIMO systems.

Abstract

Utilization of inter-base station cooperation for information processing has shown great potential in enhancing the overall quality of communication services (QoS) in wireless communication networks. Nevertheless, such cooperations require the knowledge of channel state information (CSI) at base stations (BSs), which is assumed to be perfectly known. However, CSI errors are inevitable in practice which necessitates beamforming techniques that can achieve robust performance in the presence of channel estimation errors. Existing approaches relax the robust beamforming design problems into semidefinite programming (SDP), which can only achieve a solution that is far from being optimal. To this end, this paper views robust beamforming design problems from a bilevel optimization perspective. In particular, we focus on maximizing the worst-case weighted sum-rate (WSR) in the downlink multi-cell multi-user multiple-input single-output (MISO) system considering bounded CSI errors. We first reformulate this problem into a bilevel optimization problem and then develop an efficient algorithm based on the cutting plane method. A distributed optimization algorithm has also been developed to facilitate the parallel processing in practical settings. Numerical results are provided to confirm the effectiveness of the proposed algorithm in terms of performance and complexity, particularly in the presence of CSI uncertainties.

Robust Beamforming for Downlink Multi-Cell Systems: A Bilevel Optimization Perspective

TL;DR

This work tackles robust downlink beamforming in multi-cell MU-MISO under bounded CSI errors by reformulating the worst-case weighted sum-rate maximization as a bilevel optimization problem. It introduces BLRBF, a cutting-plane based centralized algorithm, and BLADRBF, an asynchronous distributed variant, both with convergence guarantees and no reliance on computationally intensive SDP relaxations. The approach delivers superior robustness and scalability, outperforming SDP-based methods, especially at high SNR or in large-scale networks, while enabling parallel processing across base stations. The results suggest a practical, scalable pathway for robust inter-cell cooperative beamforming in realistic CSI-uncertain environments, with potential extensions to MIMO systems.

Abstract

Utilization of inter-base station cooperation for information processing has shown great potential in enhancing the overall quality of communication services (QoS) in wireless communication networks. Nevertheless, such cooperations require the knowledge of channel state information (CSI) at base stations (BSs), which is assumed to be perfectly known. However, CSI errors are inevitable in practice which necessitates beamforming techniques that can achieve robust performance in the presence of channel estimation errors. Existing approaches relax the robust beamforming design problems into semidefinite programming (SDP), which can only achieve a solution that is far from being optimal. To this end, this paper views robust beamforming design problems from a bilevel optimization perspective. In particular, we focus on maximizing the worst-case weighted sum-rate (WSR) in the downlink multi-cell multi-user multiple-input single-output (MISO) system considering bounded CSI errors. We first reformulate this problem into a bilevel optimization problem and then develop an efficient algorithm based on the cutting plane method. A distributed optimization algorithm has also been developed to facilitate the parallel processing in practical settings. Numerical results are provided to confirm the effectiveness of the proposed algorithm in terms of performance and complexity, particularly in the presence of CSI uncertainties.
Paper Structure (8 sections, 4 theorems, 25 equations, 5 figures, 2 algorithms)

This paper contains 8 sections, 4 theorems, 25 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. BLRBF and BLADRBF Methods
  4. BLRBF: Bilevel Based Robust Beamforming
  5. BLADRBF
  6. Experiments
  7. Conclusion
  8. Acknowledgments

Key Result

Theorem 1

The function $g(\boldsymbol{V}, \boldsymbol{\Delta})$ is convex with respect to $(\boldsymbol{V}, \boldsymbol{\Delta})$ and the feasible region $\mathcal{S}$ is convex when $K=1$.

Figures (5)

  • Figure 1: local feasible regions $\mathcal{S}_{i}$, global feasible region $\mathcal{S}$, and cutting planes $\mathcal{D}_{i}$ that are generated by $\text{BS}_{i}$.
  • Figure 2: Comparing the worst-case weighted sum-rates yielded by BLRBF, SDP-based beamforming and WMMSE algorithms for $M=N=K=2$.
  • Figure 3: Comparing the worst-case weighted sum-rates yielded by BLRBF, SDP-based beamforming and WMMSE algorithms for $M=K=3, N=4$.
  • Figure 4: The worst-case weighted sum-rates yielded by BLRBF algorithms for $M=4, K=10, N=64$.
  • Figure 5: Comparing the convergence rate of BLRBF and BLADRBF algorithms for $M=3, K=3, N=4$.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Proposition 1
  • Theorem 3