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Distributed Robust Multi-Cell Coordinated Beamforming with Imperfect CSI: An ADMM Approach

Chao Shen, Tsung-Hui Chang, Kun-Yu Wang, Zhengding Qiu, Chong-Yung Chi

arXiv:1107.2018v1cs.IT

TL;DR

The paper studies robust multi-cell coordinated beamforming with imperfect CSI, minimizing weighted sum power under worst-case SINR constraints while seeking decentralized operation with local CSI and limited backhaul signaling. It develops an SDR-based convex approximation and an ADMM-based distributed algorithm, and reports convergence to the centralized global optimum with lower backhaul signaling overhead than existing methods.

  • Problem

    Imperfect CSI makes robust MCBF necessary, while worst-case SINR-constrained design is difficult even centrally and decentralized solutions must use local CSI with limited backhaul signaling.

  • Method

    The paper combines semidefinite relaxation for a convex centralized approximation with ADMM to solve the formulation in a decentralized fashion.

  • Results

    The distributed robust MCBF algorithm is proven to converge to the global optimum of the centralized problem and to require less backhaul signaling than existing methods.

  • Takeaways & Limitations

    The proposed approach provides a distributed robust MCBF solution using local processing and reduced backhaul signaling while retaining the centralized optimum.

Abstract

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Multi-cell coordinated beamforming (MCBF), where multiple base stations (BSs) collaborate with each other in the beamforming design for mitigating the inter-cell interference, has been a subject drawing great attention recently. Most MCBF designs assume perfect channel state information (CSI) of mobile stations (MSs); however CSI errors are inevitable at the BSs in practice. Assuming elliptically bounded CSI errors, this paper studies the robust MCBF design problem that minimizes the weighted sum power of BSs subject to worst-case signal-to-interference-plus-noise ratio (SINR) constraints on the MSs. Our goal is to devise a distributed optimization method that can obtain the worst-case robust beamforming solutions in a decentralized fashion, with only local CSI used at each BS and little backhaul signaling for message exchange between BSs. However, the considered problem is difficult to handle even in the centralized form. We first propose an efficient approximation method in the centralized form, based on the semidefinite relaxation (SDR) technique. To obtain the robust beamforming solution in a decentralized fashion, we further propose a distributed robust MCBF algorithm, using a distributed convex optimization technique known as alternating direction method of multipliers (ADMM). We analytically show the convergence of the proposed distributed robust MCBF algorithm to the optimal centralized solution and its better bandwidth efficiency in backhaul signaling over the existing dual decomposition based algorithms. Simulation results are presented to examine the effectiveness of the proposed SDR method and the distributed robust MCBF algorithm.

EDICS: SAM-BEAM, MSP-APPL, MSP-CODR, SPC-APPL

The paper addresses robust multi-cell coordinated beamforming under imperfect CSI, targeting decentralized optimization with local CSI and reduced backhaul signaling. It combines SDR for centralized approximation with ADMM for distributed solution of the robust design.

  • Motivation: MCBF coordinates multiple base stations’ beamforming to mitigate inter-cell interference, but practical designs must account for imperfect and difficult-to-obtain inter-cell CSI.CSI errors can degrade performance and prevent mobile stations’ QoS requirements from being guaranteed.
  • Problem formulation: The target problem minimizes base stations’ weighted sum power subject to worst-case SINR requirements for mobile stations.Its worst-case constraints are effectively infinitely many nonconvex constraints and include CSI errors in desired-signal, intra-cell, and inter-cell interference terms.
  • Centralized approximation: The proposed centralized approximation applies SDR, decomposes worst-case constraints, and uses the S-lemma to reformulate them as LMIs in a convex SDP.The resulting semidefinite program can be solved efficiently by interior-point methods.
  • Centralized approximation: Under identified conditions, including one mobile station per cell or sufficiently small CSI errors, the SDR method yields the global optimum of the original robust problem.The paper also states that the SDR problem can attain the global optimum of the robust MCBF problem.
  • Distributed optimization: ADMM produces a distributed robust MCBF algorithm that converges to the global optimum of the centralized problem while using local decomposition across base stations.Slack variables representing worst-case inter-cell-interference powers reduce the messages exchanged between base stations.

II. SIGNAL MODEL AND PROBLEM STATEMENT

The paper models a multi-cell downlink in which base stations jointly design beamformers to minimize weighted transmit power while meeting users’ SINR requirements. The model initially assumes each mobile station is served by one base station and uses common-frequency transmission with single-user detection.

  • The system has Nc cells, each with one BS equipped with Nt antennas and K single-antenna MSs.
  • Each BS transmits information streams to its users using beamforming over a common frequency band.
  • The primary scenario serves each MS from only one BS, while multi-BS service is treated as an extension.
  • The received signal contains the desired signal, intra-cell interference, inter-cell interference, and additive noise.
  • The MCBF objective minimizes weighted sum power while satisfying every MS’s target SINR requirement.
  • The baseline MCBF problem can be reformulated as a convex SOCP and solved efficiently with standard convex solvers.

B. Worst-Case Robust MCBF Design

The paper formulates worst-case robust MCBF under elliptically bounded CSI errors, then uses SDR and the S-lemma to convert the difficult robust problem into a finite convex SDP approximation. This approach targets guaranteed SINR under all admissible errors while improving power efficiency over a restrictive approximation.

  • Perfect-CSI MCBF may violate users’ SINR requirements and cause outage when channel estimates contain errors.
  • In the motivating example, the non-robust design often falls below the 20 dB target, with worst-case SINR below 5 dB.
  • The robust design requires each SINR target to hold for all CSI errors inside an elliptically bounded uncertainty set.
  • SDR formulation: The proposed SDR method is reported to be more power efficient than the restrictive approximation in.
  • SDR formulation: SDR replaces rank-one beamforming matrices with general positive semidefinite matrices, linearizing the nonconvex SINR formulation.
  • SDR formulation: The S-lemma converts the infinitely many uncertainty constraints into finitely many linear matrix inequalities, yielding a convex SDP.

B. Optimality Conditions

The paper identifies conditions under which the SDR relaxation is exact and produces rank-one solutions, thereby recovering optimal beamformers for the original robust problem. Outside these conditions, rank-one optimality is not theoretically established and approximation may be required.

  • Rank-one SDR solutions are guaranteed when each cell has only one mobile station.
  • Under the stated exactness conditions, solving the SDR yields an optimal solution of the original worst-case robust MCBF problem.
  • Rank-one solutions are guaranteed for general K when intra-cell CSI is perfect and only inter-cell CSI is uncertain.
  • If both intra-cell and inter-cell CSI are uncertain, rank-one solutions are guaranteed when the CSI errors are sufficiently small.
  • For general setups, rank-one solvability is unknown theoretically, so Gaussian randomization can produce an approximate rank-one solution.
  • Simulation tests found rank-one SDR solutions for the examined problem instances, although the underlying reason remains an open research issue.

IV. DISTRIBUTED ROBUST MCBF ALGORITHM USING ADMM

The paper distributes the SDR-based robust MCBF computation across base stations using ADMM and local CSI, addressing limitations of dual decomposition. ADMM alternates primal subproblem updates with a dual update and converges to an optimal centralized solution under stated convexity and regularity conditions.

  • A centralized SDR solution requires a control center with all users’ CSI, whereas the distributed design uses only local CSI at each BS.
  • Dual decomposition is unsuitable because its decomposed problems lack strict convexity and may be unbounded below.
  • The proposed method applies ADMM to distribute optimization of the SDR problem across base stations.
  • ADMM procedure: ADMM reformulates the problem with a penalty-augmented objective while retaining equivalence to the original constrained problem.
  • ADMM procedure: ADMM solves one Gauss-Seidel iteration and one outer subgradient update per iteration to accelerate convergence.
  • Convergence: Under boundedness or invertibility assumptions, every limit point of the ADMM iterates is an optimal solution of the original problem.

B. Applying ADMM to Problem (16)

The paper reformulates the coupled SDR problem so ADMM steps become decomposable across base stations, enabling distributed robust beamforming with local CSI and limited backhaul exchange. The algorithm converges to the global optimum of the SDR problem, while reducing signaling relative to dual-decomposition approaches.

  • The original SDR formulation is difficult to distribute because its constraints are intricately coupled.
  • Introducing local and public interference variables decomposes the constraints into independent convex sets for each base station.The construction uses local ICI variables, a public ICI vector, and linear mappings between them.
  • ADMM augments the reformulated problem with penalty terms that resolve the numerical unbounded-below issue affecting conventional dual decomposition.The conventional dual inner minimization can be unbounded below, especially when the number of cells exceeds two.
  • Each base station independently solves its local beamforming problem using local CSI, broadcasts its local ICI vector, and updates shared and dual variables.The distributed steps include local optimization, backhaul exchange of t_n, public-variable computation, and independent dual updates.
  • Algorithm 2 converges to the global optimum of the SDR problem, with the resulting beamforming matrices globally optimal at convergence.The convergence result is stated through primal and dual convergence, after which the local beamforming solution is globally optimal.
  • For six cells, the proposed algorithm requires about 60% of the backhaul signaling used by the comparison algorithms.The paper attributes the savings to exchanging the local ICI vector rather than both incoming and outgoing ICI variables.
  • ADMM can be interpreted as adaptive ICI regularization that drives the base stations toward consensus on induced ICI powers.At convergence, consensus on ICI powers implies globally optimal beamforming solutions.
  • Because ADMM operates in the dual domain, an additional primal step may be needed if the algorithm stops before reaching a reasonable ICI consensus.The correction step is guaranteed feasible when the corresponding local problem is feasible for every base station.

V. EXTENSION TO FULLY COORDINATED BSS

The paper extends robust coordinated beamforming to cell-edge users jointly served by multiple base stations. It applies the SDR and S-lemma framework, with global optimality guaranteed when CSI errors are sufficiently small.

  • The extension considers cell-edge mobile stations that are simultaneously served by multiple fully coordinated base stations.The coordinated setting requires shared data streams and CSI for the cell-edge users.
  • The transmit model adds beamforming vectors for shared cell-edge data streams alongside the intra-cell transmission signals.Each base station transmits both its intra-cell signal and coordinated data streams for the cell-edge users.
  • The design objective remains to find beamforming vectors robust against possible CSI errors under worst-case constraints.
  • The SDR and S-lemma method transforms the worst-case constraints into a finite set of linear matrix inequalities and an SDP.The formulation introduces slack variables for the transformed semidefinite program.
  • If CSI errors are sufficiently small, the SDR formulation can attain the global optimum of the original worst-case robust problem.The condition is expressed through feasibility of the SDR and a positive optimal objective value.
  • A distributed algorithm for the extended SDP can also be developed using ADMM, following the same approach as for the original problem.A simulation is proposed to demonstrate guaranteed QoS for cell-edge users.

VI. SIMULATION RESULTS

The simulation section evaluates the proposed robust MCBF design and distributed optimization algorithm, and also examines the robust fully coordinated beamforming design from the preceding section.

  • The simulations examine the performance of the proposed robust MCBF design and distributed optimization algorithm.
  • The experiments also evaluate the robust fully coordinated beamforming design introduced in the preceding section.

A. Simulation Setting

The simulations model multi-cell channels with small- and large-scale fading, shadowing, path loss, antenna gains, bounded CSI errors, common SINR targets, and per-base-station power limits. Large-scale fading is assumed known, while only small-scale CSI errors remain uncertain.

  • The channel model includes both small-scale fading and large-scale effects such as shadowing and path loss.
  • The simulations assume base stations accurately track large-scale fading and experience only small-scale CSI errors.
  • Users are randomly located within cells, with serving-base-station distance at least 35 meters and inter-base-station distance of 500 meters.
  • Shadowing follows a zero-mean log-normal distribution with standard deviation 8, while preassumed CSI entries are i.i.d. complex Gaussian with zero mean and unit variance.
  • All users share noise power spectral density -162 dBm/Hz, equivalent to -92 dBm over 10 MHz, and each base station has a 46 dBm maximum power limit.
  • All users use the same SINR requirement γ, every link has antenna gain 15 dBi, and equal power weights produce a sum-power objective.
  • CSI uncertainty uses a spherical error model with common error radius ε unless otherwise specified.

B. Performance Comparison with Existing Methods

The proposed robust MCBF design improves feasibility and power efficiency over existing and non-robust alternatives, while the distributed algorithm approaches centralized performance with iteration-dependent convergence.

  • Robust MCBF achieves a much higher feasibility rate than SCBF by exploiting the degrees of freedom provided by multiple BSs.
  • The SDR problem yields rank-one solutions in all simulation tests, so its feasibility rate equals that of the original robust MCBF problem.
  • Robust designs require higher average transmission power than non-robust designs as a price for worst-case performance guarantees.
  • At γ = 10 dB, the proposed SDR method consumes around 24 dBm, whereas the method in requires 29 dBm.
  • Algorithm 2 produces near-optimal centralized solutions within 10–20 iterations for two-cell settings and about 25 iterations for three cells.
  • Convergence slows as cell or MS counts, CSI error radius, or SINR requirements increase; eight-cell settings require at least 100 iterations.

D. Performance of Robust Fully Coordinated BF

The simulations compare robust fully coordinated beamforming with robust MCBF, focusing on cell-edge users in a three-cell setting. Fully coordinated beamforming is reported as more feasible and about 3 dB more power efficient, while the SDR approximation is exact on tested instances.

  • Simulation setup: The simulation uses three cells with two mobile stations per cell, separating intra-cell and cell-edge regions.The inter-BS distance is 500 meters, and the intra-cell region has radius 235 meters.
  • Simulation setup: Fully coordinated beamforming simultaneously serves the three cell-edge mobile stations through all three base stations.In this configuration, the fully coordinated formulation uses K = 1 and L = 3.
  • SDR effectiveness: Across 17,000 channel realizations, the SDR formulation always produced rank-one solutions for the tested problem instances.Therefore, the obtained solutions were exactly optimal for the fully coordinated formulation in those instances.
  • Performance comparison: Fully coordinated beamforming was more feasible and about 3 dB more power efficient than robust MCBF when serving cell-edge mobile stations.The comparison is reported between the robust fully coordinated design and robust MCBF.
  • SDR optimality: The SDR method can yield the global optimum when each cell has one mobile station or CSI errors are sufficiently small.This condition is identified as an optimality result for the original robust problem.

APPENDIX A

Appendix A establishes rank-one properties of the SDR solution through KKT-based arguments across several cases. The proof includes special handling for single-user and zero-error conditions and connects one case to prior single-cell robust beamforming results.

  • Proof of Proposition 1: The appendix proves the relevant SDR matrix has rank one in case C1.The proof derives this from the KKT conditions and concludes that case C1 is established.
  • Proof of Proposition 1: Case C2 follows similar KKT derivations while setting Q_nnk = ∞I_Nt, corresponding to e_nnk = 0.The appendix treats this as the zero-error condition for all n and k.
  • Proof of Proposition 1: Case C3 generalizes a prior single-cell result on SDR tightness for worst-case robust beamforming.The cited single-cell setting has Nc = 1, and the appendix states that the proof follows the same idea.
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