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Max-Min Fairness in IRS-Aided Multi-Cell MISO Systems with Joint Transmit and Reflective Beamforming

Hailiang Xie, Jie Xu, Ya-Feng Liu

arXiv:2003.00906v2eess.SP

TL;DR

The paper jointly optimizes transmit and reflective beamforming to maximize minimum weighted received SINR. Its proposed designs achieve increased min-SINR performance, while inexact alternating optimization also improves on the exact design in achieved min-weighted-SINR and computational complexity.

  • Problem

    The paper jointly optimizes coordinated transmit beamforming and IRS reflective beamforming to maximize the minimum weighted received SINR at users.

  • Method

    The reflective beamforming optimization is formulated as a convex semi-definite program that can be solved to obtain an efficient solution.

  • Results

    The proposed three designs achieve significantly increased min-SINR performance against benchmark schemes, and inexact designs outperform exact alternating optimization in achieved min-weighted-SINR and computational complexity.

  • Takeaways & Limitations

    The results support using inexact alternating-optimization designs when both achieved min-weighted-SINR and computational complexity matter.

  • Takeaways & Limitations

    The paper notes that important issues remain due to space limitations, including limited resolutions and amplitude and phase considerations.

Abstract

from arXiv · show

This paper investigates an intelligent reflecting surface (IRS)-aided multi-cell multiple-input single-output (MISO) system with several multi-antenna base stations (BSs) each communicating with a single-antenna user, in which an IRS is dedicatedly deployed for assisting the wireless transmission and suppressing the inter-cell interference. Under this setup, we jointly optimize the coordinated transmit beamforming at the BSs and the reflective beamforming at the IRS, for the purpose of maximizing the minimum weighted signal-to-interference-plus-noise ratio (SINR) at the users, subject to the individual maximum transmit power at the BSs and the reflection constraints at the IRS. To solve the non-convex problem, we first present an exact-alternating-optimization design to optimize the transmit and reflective beamforming vectors in an alternating manner, in which the transmit and reflective beamforming optimization subproblems are solved exactly by using the technique of semi-definite relaxation (SDR). However, it has high computational complexity and may lead to compromised performance due to the uncertainty of randomization in SDR. To avoid these drawbacks, we further propose an inexact-alternating-optimization design, in which the transmit and reflective beamforming optimization subproblems are solved inexactly based on the principle of successive convex approximation (SCA). In addition, to further reduce the complexity, we propose a low-complexity inexact-alternating-optimization design, in which the reflective beamforming optimization subproblem is solved more inexactly. Numerical results show that the significant performance gains achieved by the proposed three designs against benchmark schemes. Moreover, the inexact-alternating-optimization designs outperform the exact-alternating-optimization one in terms of both the achieved min-weighted-SINR value and the computational complexity.

I. INTRODUCTION

The paper addresses interference in IRS-aided multi-cell MISO systems by jointly designing coordinated BS transmit beamforming and IRS reflective beamforming to maximize minimum weighted SINR. It develops exact and inexact alternating-optimization designs, showing that the inexact variants improve the performance-complexity trade-off.

  • Motivation: Inter-cell co-channel interference becomes severe in dense 5G-and-beyond networks, motivating coordinated beamforming and IRS-assisted interference management.Prior approaches include CSI-sharing coordinated transmit/receive beamforming and network MIMO.
  • Motivation: IRSs can enhance desired signals or suppress interference by coherently or destructively combining reflected signals through controllable phase shifts.The IRS is passive and is presented as a green, cost-effective way to improve wireless transmission.
  • Research gap: Prior IRS studies largely focused on single-cell settings, motivating this paper’s focus on IRS-assisted interfering multi-cell communications.Earlier work considered point-to-point, multiuser, OFDM, NOMA, and SWIPT systems, but the cited works focused on single-cell setups.
  • Problem formulation: The paper jointly optimizes coordinated BS transmit beamforming and IRS reflective beamforming to maximize minimum weighted received SINR under BS power and IRS reflection constraints.The coupling between the two beamforming vectors makes the resulting minimum-SINR maximization problem highly non-convex.
  • Proposed designs: The exact design alternates between subproblems solved using feasibility SOCPs, bisection, and SDR, whereas the inexact designs use SCA-based approximate updates.The low-complexity variant solves the reflective beamforming subproblem more inexactly to further reduce complexity.
  • Results: The three designs outperform benchmarks without an IRS or with random reflective beamforming, while inexact designs outperform the exact design in min-weighted-SINR and computational complexity.The low-complexity inexact design further reduces complexity with slightly compromised performance, and the proposed design extends to unit-amplitude reflection constraints with negligible performance loss.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models an IRS-aided multi-cell MISO system in which the IRS assists transmission and suppresses inter-cell interference. It formulates max-min weighted-SINR beamforming under BS power and IRS reflection constraints, with the IRS coupling making the problem highly non-convex.

  • System model: An IRS is deployed at the cell boundary to assist multi-cell communication and suppress inter-cell interference, especially for cell-edge users.
  • System model: The system uses quasi-static narrow-band channels, multiple multi-antenna BSs, single-antenna users, and an IRS with N > 1 reflecting units.
  • Problem formulation: The objective maximizes the minimum weighted SINR by jointly optimizing BS transmit beamforming and IRS reflective beamforming under individual BS power and IRS reflection constraints.
  • Problem formulation: The IRS introduces additional constraints and couples reflective and transmit beamforming within the SINR expressions, making the optimization highly nonlinear and non-convex.

III. EXACT ALTERNATING OPTIMIZATION

The exact-alternating-optimization approach alternates between transmit and reflective beamforming while exactly solving each subproblem with the other variable fixed.

  • Exact alternating optimization: The approach alternates optimization of the BS transmit beamforming vectors and IRS reflective beamforming vector.
  • Exact alternating optimization: At each iteration, “exact” means that the corresponding transmit and reflective beamforming subproblems are solved exactly with the other beamformer fixed.

A. Coordinated Transmit Beamforming Optimization

For fixed IRS reflection, the transmit-beamforming subproblem is transformed into feasibility problems and solved optimally using SOCP with bisection over the target SINR.

  • Problem transformation: With the reflective beamforming vector fixed, the coordinated transmit-beamforming problem remains non-convex and is reformulated by fixing the auxiliary target t.
  • Problem transformation: Feasibility of the fixed-t problem determines whether t is below or above the optimum, enabling bisection search over t.
  • SOCP solution: The resulting transmit-beamforming formulation is an SOCP that standard convex solvers can solve optimally.

B. Reflective Beamforming Optimization

For fixed transmit beamforming, the reflective-beamforming subproblem is lifted into an SDP through semidefinite relaxation and reconstructed using rank-one recovery or Gaussian randomization. The resulting exact alternating method has high complexity and can suffer from randomization-dependent performance.

  • Reflective beamforming formulation: The reflective-beamforming problem is lifted by defining V = ¯v¯v^H and relaxing its non-convex rank constraint.
  • SDR solution: The relaxed feasibility problem is a convex SDP that can be solved optimally, after which a rank-one reflective solution is reconstructed.
  • Rank-one recovery: If the relaxed solution has rank greater than one, Gaussian randomization generates candidate rank-one solutions and selects the best one.
  • Limitations: The exact alternating approach may have compromised performance because of randomization uncertainty and has very high computational complexity from repeatedly solving feasibility subproblems.

IV. INEXACT ALTERNATING OPTIMIZATION

The inexact-alternating-optimization design addresses the drawbacks of exact alternating optimization by updating transmit and reflective beamforming to increase the min-weighted-SINR at each iteration.

  • IV. INEXACT ALTERNATING OPTIMIZATION: The design alternately updates coordinated transmit and reflective beamforming vectors without exactly solving both original subproblems.Each iteration seeks updated beamforming vectors that increase the min-weighted-SINR value.
  • IV. INEXACT ALTERNATING OPTIMIZATION: At iteration l, the method uses the previous local point of the beamforming vectors as the basis for alternating updates.The previous iteration provides the local point for constructing the next transmit and reflective beamforming updates.

A. Inexact Coordinated Transmit Beamforming Update

The inexact transmit-beamforming update solves a convex approximation that produces a feasible update with non-decreasing min-weighted-SINR, avoiding the exact subproblem’s bisection search.

  • A. Inexact Coordinated Transmit Beamforming Update: The transmit-beamforming vectors are updated under fixed reflective beamforming using an inexact optimization problem based on the current iteration point.The update is formed from the previous reflective vector and current transmit-beamforming point.
  • A. Inexact Coordinated Transmit Beamforming Update: A phase choice makes the relevant desired-signal term non-negative, allowing the transmit update to be transformed into an SOCP.The resulting problem can be solved optimally using standard convex optimization methods.
  • A. Inexact Coordinated Transmit Beamforming Update: Solving problem (P4) once yields a feasible coordinated transmit beamforming update with a non-decreasing min-weighted-SINR value.The original transmit-beamforming problem is not solved exactly in this design.
  • A. Inexact Coordinated Transmit Beamforming Update: The inexact transmit update avoids the bisection search required by exact alternating optimization and can provide superior performance with inexact reflective updates.The claimed benefits are reduced computational complexity and improved performance relative to the exact approach.

B. Inexact Reflective Beamforming Update

The inexact reflective-beamforming update applies SCA to obtain a convex problem whose solution preserves non-decreasing min-weighted-SINR and supports convergence of the alternating design.

  • B. Inexact Reflective Beamforming Update: The reflective beamforming vector is updated to increase the min-weighted-SINR after the transmit-beamforming update.The update is performed under the current coordinated transmit beamforming.
  • B. Inexact Reflective Beamforming Update: Because the reflective-beamforming subproblem remains non-convex, SCA replaces a convex term with its first-order Taylor expansion.The Taylor expansion provides a lower bound at the current point and yields a convex approximation.
  • B. Inexact Reflective Beamforming Update: Introducing an auxiliary variable transforms the approximation into problem (P5.1), a convex problem solved optimally by CVX.The previous reflective vector with zero auxiliary variable is feasible for the approximated problem.
  • B. Inexact Reflective Beamforming Update: Each iteration solves two convex optimization problems, giving the inexact design lower computational complexity than the exact alternating approach.The two problems update coordinated transmit beamforming and reflective beamforming, respectively.

V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION

The low-complexity design further reduces computation by solving the reflective-beamforming approximation with subgradient projection rather than CVX, while retaining monotonicity and convergence.

  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The low-complexity design solves the reflective-beamforming subproblem more inexactly to reduce computational complexity.The coordinated transmit-beamforming update remains inexact, while the reflective update uses a lower-cost method.
  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The reflective-beamforming problem is reformulated as a constrained convex optimization problem and solved with subgradient projection.The feasible set constrains each reflection coefficient magnitude to at most one, and projection exploits separable reflection constraints.
  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The method uses subgradients of the maximum objective and projects each iterative update onto the feasible reflection-coefficient set.A constant step length is used, and the best point found across the subgradient iterations is retained.
  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The subgradient projection update is not generally a descent method, so the algorithm tracks the best point found so far.Numerical results report that a constant step-length rule reaches the best objective value more efficiently than alternatives.
  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The reflective-beamforming update has complexity O(KNT), lower than the interior-point solution used in the inexact design.Here, K is associated with users, N with IRS elements, and T with the number of subgradient iterations in the cited complexity expression.
  • V. LOW-COMPLEXITY INEXACT ALTERNATING OPTIMIZATION: The low-complexity approach is cheaper than the inexact alternating design and remains convergent because its min-weighted-SINR is non-decreasing after each update.The resulting method is presented as Algorithm 3.

VI. NUMERICAL RESULTS

Numerical evaluations show that the proposed alternating-optimization designs improve minimum weighted SINR over benchmark schemes, while inexact designs generally offer better performance-complexity trade-offs than exact SDR-based optimization.

  • Randomization and convergence: The inexact and low-complexity inexact algorithms increase minimum SINR monotonically over iterations and converge, whereas exact optimization terminates at a lower value.The comparison is reported for the setting Pmax = 35 dBm.
  • Randomization and convergence: The exact-alternating-optimization method can achieve lower minimum SINR because Gaussian randomization introduces uncertainty during iterations.Its achieved minimum SINR increases with more randomizations, motivating 1000 randomizations for balancing performance and complexity.
  • IRS effects: Proper reflective beamforming is necessary to realize the IRS benefit, since random reflective beamforming performs similarly to the no-IRS benchmark.With unit-amplitude reflection constraints, the proposed algorithms retain performance close to the unconstrained upper bound, with negligible loss.
  • Complexity and operating conditions: The proposed designs become more advantageous than exact optimization in high-power settings, while low-complexity inexact optimization uses the least CPU time with slightly compromised performance.The low-complexity design has complexity similar to ZF and MRT benchmarks but achieves much better performance.
  • Complexity and operating conditions: Minimum SINR increases with the number of reflecting units, but the performance gap between exact and inexact optimization also becomes larger.The proposed approaches’ gains over benchmarks decrease as users move toward cell centers and direct inter-cell interference becomes stronger.

VII. CONCLUDING REMARKS

The paper formulates IRS-assisted multi-cell MISO beamforming as a minimum weighted-SINR maximization problem and proposes three alternating-optimization algorithms with different performance-complexity trade-offs. Numerical results support IRS-assisted gains and favor inexact optimization, while the study leaves channel acquisition and practical IRS hardware issues for future work.

  • Problem and algorithms: The optimization jointly designs coordinated BS transmit beamforming and IRS reflective beamforming to maximize the minimum weighted SINR under BS power and IRS reflection constraints.The objective addresses all users in the IRS-aided multi-cell MISO system.
  • Problem and algorithms: The paper proposes exact, inexact, and low-complexity inexact alternating-optimization algorithms to balance performance and complexity.They are identified as Algorithms 1, 2, and 3, respectively.
  • Main findings: The dedicated IRS considerably improves SINR by enhancing received signal strength and suppressing inter-cell interference, especially for cell-edge users.The reported conclusion attributes the improvement to both signal enhancement and interference suppression.
  • Main findings: Inexact alternating optimization jointly optimizes transmit and reflective beamforming with reduced complexity, guaranteed convergence, and better performance than conventional exact alternating optimization.This conclusion is stated for the paper’s proposed inexact approach.
  • Limitations: The study assumes globally available perfect CSI and omits CSI signaling overhead, which may create substantial training, sharing, and latency burdens in larger systems.The authors identify channel estimation and signaling methods as an unresolved issue.
  • Limitations: Continuous, independently controlled IRS amplitudes and phases may increase fabrication and implementation complexity, motivating limited-resolution and correlated-hardware models.These hardware impairment issues are left for further investigation.
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