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Joint Base Station Clustering and Beamformer Design for Partial Coordinated Transmission in Heterogenous Networks
Mingyi Hong, Ruo-Yu Sun, Hadi Baligh, Zhi-Quan Luo
TL;DR
The paper addresses interference management in multicell MIMO HetNets, where dense BS deployments make coordination costly and general utility optimization difficult. It jointly designs user-centric BS clusters and linear beamformers through sparse optimization and an iterative group-LASSO-based procedure. The method is reported to converge to stationary or local optimal solutions and to achieve high throughput and fairness with small clusters and lower transmission power in simulations.
Problem
Dense HetNets create substantial interference and coordination overhead, while general utility maximization in MIMO interfering networks is computationally intractable.
Method
The paper jointly optimizes BS clustering and linear beamformers using a nonsmooth utility objective, an equivalent weighted-MSE reformulation, and iterative group-LASSO updates.
Results
The proposed algorithm converges to a stationary solution and simulations show high system throughput and fairness with small cluster sizes and significantly lower transmission power.
Takeaways & Limitations
A single sparsity penalty parameter provides a way to balance throughput and cluster size in partial coordinated transmission.
Abstract
from arXiv · showhide
We consider the interference management problem in a multicell MIMO heterogenous network. Within each cell there are a large number of distributed micro/pico base stations (BSs) that can be potentially coordinated for joint transmission. To reduce coordination overhead, we consider user-centric BS clustering so that each user is served by only a small number of (potentially overlapping) BSs. Thus, given the channel state information, our objective is to jointly design the BS clustering and the linear beamformers for all BSs in the network. In this paper, we formulate this problem from a {sparse optimization} perspective, and propose an efficient algorithm that is based on iteratively solving a sequence of group LASSO problems. A novel feature of the proposed algorithm is that it performs BS clustering and beamformer design jointly rather than separately as is done in the existing approaches for partial coordinated transmission. Moreover, the cluster size can be controlled by adjusting a single penalty parameter in the nonsmooth regularized utility function. The convergence of the proposed algorithm (to a local optimal solution) is guaranteed, and its effectiveness is demonstrated via extensive simulation.
I. INTRODUCTION
Heterogeneous networks improve coverage through dense BS deployments but create substantial interference and coordination overhead. The paper jointly designs user-centric BS clusters and beamformers through a sparse-optimization approach rather than separating clustering from beamformer design.
- Dense HetNets use macro, micro, and pico BSs to extend coverage for cell-edge and hotspot users, but transmitter proximity creates substantial interference.
- Coordinated transmission can mitigate interference, but utility maximization in general MIMO interfering networks is computationally intractable.
- Small BS coordination clusters reduce backhaul signaling because user data is shared only among each user’s serving BSs.
- The paper jointly optimizes possibly overlapping coordination clusters and linear beamformers in large-scale HetNets using a single-stage nonsmooth utility optimization problem.
- An equivalent regularized weighted MSE formulation enables an iterative algorithm whose computation is separable across cells and convex across variable blocks.
- The proposed iterative algorithm converges to a stationary solution, with effectiveness evaluated through extensive simulations.
II. SYSTEM MODEL AND PROBLEM FORMULATION
The system models a multicell HetNet in which users are jointly served by selected intra-cell BS subsets while inter-cell coordination uses beamforming. The formulation encodes small serving subsets as group sparsity and penalizes the utility accordingly.
- Each cell contains distributed BSs connected to a central controller, and each user is jointly served by a subset of that cell’s BSs.
- Partial joint processing restricts each user’s serving set while targeting high system throughput and/or fairness.
- A small serving set corresponds to a beamformer with only a few nonzero BS blocks, producing a group-sparse structure.
- The mixed ℓ2/ℓ1 norm promotes group sparsity by applying an ℓ1 norm to the vector of BS-block ℓ2 norms, yielding a group LASSO problem.
- The nonsmooth penalized utility maximization jointly designs linear beamformers and controls sparsity through cell-specific parameters λk.
- Direct solution is challenging because the unpenalized problem is NP-hard for many common utilities, while standard group LASSO methods assume convex quadratic smooth objectives.
III. EQUIVALENT FORMULATION
The paper reformulates the nonsmooth utility maximization as a regularized weighted MSE minimization problem to obtain a more tractable optimization structure.
- A general equivalence relationship transforms the utility maximization problem into a regularized weighted MSE minimization problem.
- The reformulation is designed for the nonsmooth setting and facilitates development of an efficient iterative algorithm.
- The transformation is presented as a generalization of an earlier equivalence result for smooth utility functions.
A. Single User Per Cell with Sum Rate Utility
For a single scheduled user per cell and sum-rate utility, the paper introduces a sparse beamforming formulation and an equivalent weighted-MSE problem. Stationarity is preserved between the two formulations, and their global optima coincide.
- The single-user-per-cell case represents one orthogonally scheduled resource and uses sum rate as the system utility.
- The formulation uses beamformers, channels, rates, and MSE-related variables for the single user in each cell.
- Introducing weight variables yields a regularized weighted MSE minimization problem alongside the sparse beamforming formulation.
- With beamformers and receivers fixed, each weight has a closed-form solution wk = 1/ek.
- Stationary solutions of the regularized weighted-MSE problem and the original sparse utility problem correspond in both directions.
- The global optimal beamformers of the two problems are identical.
B. Multiple Users Per Cell with More General Utility
The paper extends its equivalence framework to multiple users per cell and more general utility functions satisfying two stated conditions. Under these conditions, stationary solutions of the original nonsmooth utility problem correspond to stationary solutions of a regularized weighted MSE problem, with identical global optima.
- The original problem is transformed into a regularized weighted MSE minimization problem for multiple users per cell.
- The utility function must be concave and strictly increasing, while u_ik(-log(x)) must be strictly convex for 0 ≤ x ≤ 1.
- The admissible utility family includes weighted sum rate and geometric mean of one plus rates.
- Stationary solutions of the regularized problem map to stationary solutions of the original problem, and the converse also holds.
- The two problems have identical global optimal beamformers.
IV. JOINT CLUSTERING AND BEAMFORMER DESIGN
The algorithm solves the regularized weighted sum-MSE formulation using block coordinate descent, exploiting its equivalence to the original nonsmooth utility problem.
- Block coordinate descent is applied to the regularized weighted sum-MSE problem because it is equivalent to the original nonsmooth utility maximization problem.
A. The Algorithm
The algorithm alternates updates of receivers, weights, and beamformers, with the beamformer step decomposed into cell-wise convex group-LASSO subproblems. A sparsity condition removes beamformer groups, and the resulting S-WMMSE algorithm converges to a stationary solution.
- With fixed beamformers and weights, the receiver update is the MMSE receiver; with fixed beamformers and receivers, the weight update has a closed form.
- With receivers and weights fixed, the beamformer problem decomposes into K independent convex problems, one for each cell.
- Each cell-wise beamformer update is a quadratically constrained group-LASSO problem, requiring a customized solver because of the sum-power constraint.
- The beamformer update solves a primal-dual system using subdifferentials, active-user sets, and bisection over the power-budget multiplier.
- A beamformer group is forced to zero whenever the relevant norm falls below the sparsity threshold, producing the desired group-sparse structure.
- In the single-user, single-antenna case, the beamformer update simplifies to a closed-form procedure without bisection.
- The S-WMMSE algorithm converges to a stationary solution of the original nonsmooth utility maximization problem.
B. Parameter Selection
The implementation guidelines select initial bounds for the bisection variables using monotonicity and feasibility properties, supporting the multiplier and auxiliary-variable searches.
- The outer bisection searches the power-budget multiplier, while the inner loop searches the auxiliary variable for each active user.
- The initial multiplier lower bound is zero, and an upper bound is chosen to guarantee the required feasibility condition.
- The beamformer norm is monotonically decreasing in the multiplier, enabling a bounded bisection search that satisfies feasibility and complementarity.
- The monotonicity and positive-semidefinite properties used in the bounds are stated explicitly for the relevant update expressions.
- Initial bounds for each auxiliary variable can be determined from the selected multiplier bounds.
C. Distributed Implementation
The proposed algorithm can be implemented distributively under specified channel-knowledge, user-feedback, and inter-macro-BS control-information assumptions. Clustering need not run every transmission interval because fixed clusters can support beamformer updates or remain unchanged.
- Distributed implementation: Distributed implementation requires each macro BS to know in-cell channels to all users, users to feed back information, and macro BSs to exchange control information.These are assumptions A-1 through A-3 for distributed execution.
- Distributed implementation: During each iteration, users measure received-signal covariance, update weights and receive beamformers, and feed the variables to a serving BS.
- Practical operation: Because CSI acquisition and sharing are costly, full sparse clustering can be performed only every several transmission time intervals.Between clustering updates, beamformers may be updated without regularization or fixed.
V. NUMERICAL RESULTS
Numerical experiments evaluate the proposed S-WMMSE algorithm across multicell networks, utilities, and baselines. It achieves throughput and fairness close to stronger cooperation while using moderate or smaller clusters and, in larger networks, lower transmission power.
- Cluster structure and convergence: The proposed algorithm produces overlapping clusters whose sizes are significantly reduced relative to serving every user by all BSs in a single-cell example.The convergence plots count outer iterations using an objective-value stopping criterion of |f(vt+1) − f(vt)| < 10^-1.
- Sum-rate results: The K = 4 sum-rate experiments show throughput close to full cooperation while using moderate cluster sizes, including favorable comparison with full per-cell ZF using cluster size 20.
- Penalty selection: Adaptive λk choices generate smaller clusters with similar performance to fixed-parameter choices, but the fixed-parameter convergence proof no longer applies.The adaptive algorithm usually converged in simulations.
- Proportional-fair results: In the K = 10 proportional-fair experiments, the proposed method achieves high throughput and fairness with small clusters and significantly lower transmission power.The comparison includes throughput, averaged cluster size, relative per-BS power, and individual user-rate distributions.
- Proportional-fair results: The Sparse-Neighbor scheme has similar system throughput to the original limited-cooperation comparison but produces larger cluster sizes.
VI. CONCLUDING REMARKS
The paper jointly designs BS clustering and linear beamformers through a nonsmooth sparse-optimization formulation. Its reformulations support an iterative group-LASSO-based procedure with stationary-point convergence, while simulations select few serving BSs with minor throughput or fairness loss.
- VI. CONCLUDING REMARKS: Joint BS clustering and downlink linear beamformer design is formulated as a single-stage nonsmooth utility maximization problem.
- VI. CONCLUDING REMARKS: A mixed ℓ2/ℓ1 penalty induces group sparsity in virtual-BS beamformers, selecting only a few serving BSs for each user.
- VI. CONCLUDING REMARKS: The resulting iterative algorithm uses group-LASSO updates, with closed-form computations that can be distributed to individual cells.
- VI. CONCLUDING REMARKS: Simulation results show that selecting a few serving BSs incurs minor loss in system throughput and/or user fairness.
- VI. CONCLUDING REMARKS: The algorithm converges to a stationary solution of the original nonsmooth utility maximization problem under the stated regularity and block-optimization conditions.
APPENDIX D PROOF OF THEOREM 1
The proof establishes convergence by analyzing the block-coordinate descent updates. Convexity, differentiability, separability, and unique block solutions yield global optimization of each subproblem and stationarity of the overall algorithm.
- APPENDIX D PROOF OF THEOREM 1: The S-WMMSE algorithm’s convergence proof reduces to showing convergence to a stationary solution of problem (P2).
- APPENDIX D PROOF OF THEOREM 1: The beamformer-update procedure decomposes into K independent convex subproblems, each solved globally by block-coordinate descent.
- APPENDIX D PROOF OF THEOREM 1: Block-coordinate descent converges globally for each convex nonsmooth subproblem because the nonsmooth objective is separable, the smooth part is differentiable, and each block has a unique solution.
- APPENDIX D PROOF OF THEOREM 1: Applying the block-coordinate descent convergence theorem to the three variable blocks establishes convergence of S-WMMSE to a stationary solution of problem (P2).