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Robust Beamforming for Downlink Multi-Cell Systems: A Bilevel Optimization Perspective
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Proposes BLRBF and a distributed variant BLADRBF that cast worst-case weighted sum-rate robust beamforming in multi-cell MISO downlink as a bilevel problem solved with cutting planes and augmented Lagrangian inner steps, compared against SDP-based and WMMSE beamforming.
Problem
Maximizing the worst-case weighted sum-rate of downlink multi-cell multi-user MISO networks under bounded CSI errors, where SDP relaxations are suboptimal and computationally expensive at scale.
Approach
Reformulates the robust beamforming min-max problem as a bilevel optimization (lower level finds worst-case CSI error, upper level optimizes beamformers), then solves it with a cutting-plane method (BLRBF) and an asynchronous distributed extension (BLADRBF) where BSs share cutting planes.
Key results
- BLRBF achieves higher worst-case weighted sum-rate than SDP-based beamforming, especially at high SNR, for M=K=3, N=4 (saturating at higher SNR with higher value).
- For M=4, K=10, N=64 the SDP-based method takes over 4 hours while BLRBF finishes within an acceptable time.
- BLADRBF attains the same final convergence value as BLRBF but converges faster.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BLRBF instead of SDP-based beamforming | Robust beamforming · Multi-cell MISO systems, high SNR | worst-case weighted sum rate | Helped |
| Increase the CSI uncertainty-region radius | Robust beamforming · Multi-cell MISO systems | robust weighted sum rate | Hurt |
| Use BLRBF instead of SDP-based beamforming | Robust beamforming · M=K=3, N=4 | optimized robust weighted sum rate | Helped |
| Use BLADRBF instead of BLRBF | Robust beamforming · M=3, K=3, N=4 | convergence rate | Helped |
| Use BLADRBF instead of BLRBF | Robust beamforming · M=3, K=3, N=4 | final convergence value | No clear effect |
| Use BLRBF instead of WMMSE | Beamforming with perfect CSI · Multi-cell MISO systems, ε=0 | worst-case sum rate | No clear effect |
| Use BLRBF instead of SDP-based beamforming | Beamforming with perfect CSI · Multi-cell MISO systems, ε=0 | worst-case sum rate | Helped |
| Scale the network to M=4, K=10, N=64 | SDP-based beamforming · Large-scale multi-cell MISO system | computation time | Hurt |
Limitations
- Convexity of the feasible region is proven only for K=1 (Theorem 1).
- Relies on approximation of lower-level solution via a finite number of augmented Lagrangian iterations and relaxed constraint with small epsilon.
- Experiments are limited to small simulated Rayleigh-fading setups with 10 channel realizations.
Experimental setup
Datasets: simulated Rayleigh fading channels
Benchmarks: simulated multi-cell MISO downlink (M=N=K=2, M=K=3 N=4, M=4 K=10 N=64)
Metrics: worst-case weighted sum-rate computation time convergence rate
Baselines: SDP-based robust beamforming WMMSE
Builds on: CPBO WMMSE S-Lemma
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