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Intelligent Reflecting Surface Aided Multigroup Multicast MISO Communication Systems

Gui Zhou, Cunhua Pan, Hong Ren, Kezhi Wang, Arumugam Nallanathan

arXiv:1909.04606v4eess.SPcs.IT

TL;DR

The paper addresses sum-rate maximization in IRS-assisted multigroup multicast MISO systems under coupled precoding, reflection, power, and unit-modulus constraints. It develops MM-based alternating-optimization algorithms, including SOCP-based and closed-form low-complexity updates. Simulations report spectral- and energy-efficiency benefits, convergence, and reduced complexity for the IRS-assisted system.

  • Problem

    IRS-assisted multigroup multicast design requires jointly optimizing BS precoding and IRS reflections, while multicast rates produce a non-differentiable objective and SDR methods have high complexity.

  • Method

    The paper applies MM-based alternating optimization, using concave lower-bound surrogates to obtain SOCP updates and closed-form low-complexity updates.

  • Results

    Simulations demonstrate spectral- and energy-efficiency benefits of IRS assistance and illustrate convergence and low complexity of the proposed algorithms.

  • Takeaways & Limitations

    Adjusting passive IRS reflection coefficients can enhance received signal strength while supporting efficient multigroup multicast optimization.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) has recently been envisioned to offer unprecedented massive multiple-input multiple-output (MIMO)-like gains by deploying large-scale and low-cost passive reflection elements. By adjusting the reflection coefficients, the IRS can change the phase shifts on the impinging electromagnetic waves so that it can smartly reconfigure the signal propagation environment and enhance the power of the desired received signal or suppress the interference signal. In this paper, we consider downlink multigroup multicast communication systems assisted by an IRS. We aim for maximizing the sum rate of all the multicasting groups by the joint optimization of the precoding matrix at the base station (BS) and the reflection coefficients at the IRS under both the power and unit-modulus constraint. To tackle this non-convex problem, we propose two efficient algorithms under the majorization--minimization (MM) algorithm framework. Specifically, a concave lower bound surrogate objective function of each user's rate has been derived firstly, based on which two sets of variables can be updated alternately by solving two corresponding second-order cone programming (SOCP) problems. Then, in order to reduce the computational complexity, we derive another concave lower bound function of each group's rate for each set of variables at every iteration, and obtain the closed-form solutions under these loose surrogate objective functions. Finally, the simulation results demonstrate the benefits in terms of the spectral and energy efficiency of the introduced IRS and the effectiveness in terms of the convergence and complexity of our proposed algorithms.

I. INTRODUCTION

The paper studies IRS-assisted multigroup multicast MISO transmission, where jointly designing BS precoding and IRS reflection coefficients is challenging because multicast rates and variables are coupled. It develops MM-based algorithms and reports spectral-efficiency, energy-efficiency, convergence, and complexity benefits.

  • Motivation: Multigroup multicast lets users sharing requested content receive a common stream, reducing the pressure caused by high-interference, high-complexity unicast transmission as user numbers grow.The group rate is constrained by the minimum user rate within that group.
  • Problem formulation: The paper jointly optimizes the BS precoding matrix and IRS reflection coefficients to maximize the sum rate of all multicast groups under transmit-power and unit-modulus constraints.The multicast objective is non-differentiable because each group rate is limited by its worst user, while the design variables are highly coupled.
  • Proposed methods: The proposed solution uses alternating optimization within an MM framework, replacing the non-concave objective with a biconcave surrogate and decoupling the variable updates.The first algorithm solves corresponding SOCP subproblems for the precoding matrix and reflection coefficients.
  • Proposed methods: A lower-complexity algorithm derives closed-form updates by applying a log-sum-exp lower bound and additional surrogate objectives instead of repeatedly solving high-complexity SOCPs.This approach targets the computational burden associated with the large number of precoding and reflection-coefficient parameters.
  • Results: Simulation results show that IRS assistance improves spectral and energy efficiency relative to conventional massive MIMO, while the proposed algorithms converge rapidly and have low complexity.The IRS can enhance received signal strength through adjusted passive reflection coefficients without additional transmit power.

B. Problem Formulation

The paper formulates IRS-assisted multigroup multicast beamforming as joint optimization of BS precoding and IRS reflection coefficients for maximizing group sum rate. It applies MM surrogates and alternating optimization to address the objective's non-concavity, non-differentiability, and non-convex unit-modulus constraints.

  • B. Problem Formulation: The objective maximizes the sum rate across multicast groups by jointly optimizing the precoding matrix F and reflection vector e.Each group's rate is limited by its users' rates through the multicast formulation.
  • B. Problem Formulation: Problem (8) is non-convex because its objective is non-differentiable and non-concave, while the unit-modulus constraint set for e is also non-convex.These properties make direct optimization difficult.
  • C. Majorization-Minimization Method: MM replaces the original objective with an easier surrogate function that is optimized over the feasible set.The surrogate is constructed at the current iterate while satisfying the MM conditions.
  • III. SOCP-BASED MM METHOD: The proposed SOCP-based MM method first introduces a concave surrogate for the non-convex objective and alternately updates the two variable sets.The alternating optimization decouples the precoding and reflection-variable subproblems.
  • III. SOCP-BASED MM METHOD: The resulting surrogate problem preserves the multicast pointwise-minimum structure while making the variable blocks separately concave.The concave minorization is applied at the previous iteration's solution.
  • III. SOCP-BASED MM METHOD: The user-rate surrogate is biconcave in F and e, so each variable can be optimized while holding the other fixed.This biconcavity enables alternating optimization of the two blocks.

A. Optimizing the Precoding Matrix F

With one variable block fixed at a time, the method reformulates the precoding and reflection updates using quadratic surrogates and auxiliary variables. The precoding subproblem becomes an SOCP, while the reflection update requires relaxing the unit-modulus set.

  • A. Optimizing the Precoding Matrix F: With e fixed, the minorized user-rate expression is rewritten as a quadratic function of the precoding matrix F.The formulation uses the group selection vector t_g and associated channel-dependent quantities.
  • A. Optimizing the Precoding Matrix F: Auxiliary variables γ represent the pointwise-minimum terms in the precoding subproblem.This reformulation converts the subproblem into a tractable conic form.
  • A. Optimizing the Precoding Matrix F: Problem (13) is an SOCP whose global solution can be obtained with a solver such as MOSEK.The SOCP formulation handles the power constraint and rate-related constraints.
  • A. Optimizing the Precoding Matrix F: With F fixed, the surrogate is rewritten as a function of the reflection coefficient vector e and reformulated using auxiliary variables κ.The resulting reflection subproblem remains non-convex because of the unit-modulus feasible set.
  • A. Optimizing the Precoding Matrix F: The unit-modulus set is replaced by a relaxed convex set imposing quadratic magnitude inequalities and fixing e_M+1 = 1.The relaxation uses e^Hdiag(i_m)e ≤ 1 for each element index m.
  • A. Optimizing the Precoding Matrix F: The relaxed reflection solution is converted into a locally optimal e through element-wise phase extraction and exponentiation.The phase operation is applied element-wise to the relaxed solution.

C. Algorithm development

Algorithm 1 alternates precoding and reflection updates until the original objective converges, while a second MM construction replaces SOCP subproblems with closed-form updates to reduce complexity. The SOCP algorithm has a convergence guarantee to a KKT point.

  • C. Algorithm development: Algorithm 1 initializes F0 and e0, then alternately solves the precoding and reflection SOCP subproblems until the objective converges.Each iteration updates F using the current e and then e using the updated F.
  • C. Algorithm development: Algorithm 1's computational cost mainly comes from solving the two SOCP problems for F and e.The complexity analysis identifies these two subproblems as the dominant sources.
  • C. Algorithm development: The per-iteration complexity includes terms depending on G, K, M, and the SOCP dimensions for the precoding and reflection subproblems.The expressions contain powers such as NGK^3.5 and (M+1)K^3.5.
  • C. Algorithm development: The objective sequence generated by Algorithm 1 is guaranteed to converge, and its solution converges to a KKT point.This is stated as Theorem 1.
  • C. Algorithm development: A lower-complexity algorithm is introduced because solving two SOCPs at every iteration is computationally expensive.The alternative method targets closed-form subproblem solutions.
  • C. Algorithm development: The non-differentiable objective is smoothed with a log-sum-exp lower bound controlled by a positive parameter μ_g.Larger μ_g improves approximation accuracy but can make the problem nearly ill-conditioned.
  • C. Algorithm development: The smoothed group-rate surrogate is biconcave in F and e, supporting alternating updates under the MM framework.The biconcavity is established through the concavity of the constituent function.

A. Optimizing the Precoding Matrix F

The low-complexity MM method minorizes each concave block with quadratic surrogates and obtains globally optimal block updates using a Lagrange multiplier for the power constraint. The reflection update uses a corresponding minorizer over the non-convex feasible set.

  • A. Optimizing the Precoding Matrix F: Although the group surrogate is concave and continuous in F, it remains too complex for direct optimization.The method therefore constructs a quadratic MM minorizer at the current precoding iterate.
  • A. Optimizing the Precoding Matrix F: Replacing the group objective with its quadratic minorizer yields a surrogate precoding problem.The surrogate satisfies the required MM conditions.
  • A. Optimizing the Precoding Matrix F: A nonnegative Lagrange multiplier τ is associated with the power constraint to derive the next precoding iterate.Stationarity with respect to F* gives the candidate update.
  • A. Optimizing the Precoding Matrix F: The power-constraint expression decreases with τ, enabling the multiplier to be selected according to whether the constraint holds at τ = 0.Otherwise, a positive τ is chosen so the constraint holds with equality.
  • A. Optimizing the Precoding Matrix F: With F fixed, the method constructs a minorizer for the reflection-vector subproblem while accounting for the non-convex feasible set's tangent cone.The modified MM condition supports stationarity convergence over the non-convex set.
  • A. Optimizing the Precoding Matrix F: Replacing the reflection objective with its minorizer produces a surrogate problem whose globally optimal e update is given in closed form.The update uses an element-wise exponential operation.

C. Low-complexity algorithm design

The low-complexity MM algorithm alternately updates the precoding matrix and reflection coefficients using nonlinear fixed-point maps, projection, and SQUAREM acceleration. It iterates until the objective function converges.

  • Low-complexity MM algorithm: SQUAREM accelerates alternating MM updates of the precoding matrix F and reflection coefficient vector e.The algorithm uses fixed-point maps M_F(·) and M_e(·), with lower-bound tightness affecting convergence speed.
  • Low-complexity MM algorithm: Algorithm 2 initializes F_0 and e_0, then applies fixed-point updates and projection steps to enforce the power and unit-modulus constraints.Projection is used to force iterates back into the feasible sets after extrapolated updates.
  • Low-complexity MM algorithm: The method computes two successive fixed-point iterates for each variable, forms update differences, and applies an extrapolated projected step.The F and e updates follow the same extrapolation structure before advancing the iteration counter.
  • Low-complexity MM algorithm: The algorithm terminates when the objective function F(F, e) converges.This stopping condition is applied after updating both the precoding matrix and reflection coefficient vector.

D. Complexity Analysis

The analysis characterizes the low-complexity algorithm's per-iteration cost and convergence properties, and describes simulation settings and comparison baselines. Algorithm 2 is reported to reduce complexity and CPU time relative to the SOCP-based Algorithm 1.

  • Complexity Analysis: The approximate per-iteration complexity of Algorithm 2 is O(4N^3K + 2N^2K^2 + 3MNGK + K(M + 1)^3).This expression combines the costs of the fixed-point maps for F and e while neglecting lower-order terms.
  • Complexity Analysis: Algorithm 2 has lower computational complexity and requires less CPU time than the SOCP-based Algorithm 1.The comparison is summarized in Table I.
  • Convergence Analysis: The objective values generated by the alternating-optimization procedure are monotonically increasing.The monotonicity follows from the successive surrogate-objective inequalities for the two variable updates.
  • Convergence Analysis: The precoding and reflection-coefficient sequences have limit points characterized as d-stationary and B-stationary, respectively.The distinction reflects the convex feasible set for F and the non-convex feasible set for e.
  • Convergence Analysis: Theorem 5 states that the optimal solution converges to a KKT point of Problem (24).The convergence claim is stated as a theorem in the paper.
  • Simulation Setup: Simulations average each plotted point over 100 independent trials for an IRS-aided system with BS antennas and IRS reflecting elements.The setup places the BS at (0 m, 0 m) and the IRS at (100 m, 0 m), with Rayleigh fading and specified path-loss exponents.
  • Baseline Schemes: The simulations compare IRS-Alg. 1 and IRS-Alg. 2 with no-IRS versions, 2-bit phase-resolution baselines, and a full-duplex AF relay benchmark.The 2-bit baselines address the practical difficulty of realizing continuous phase shifts.

C. Convergence of the Proposed Algorithms

The proposed algorithms converge to comparable solutions, while Algorithm 2 reaches the same objective with substantially lower CPU time. Simulations also show IRS gains in sum rate and energy efficiency under several system dimensions.

  • Initialization: The uniform-power initialization for F and all-one initialization for e performs almost identically to exhaustive random initialization.This indicates that the simple initialization is a good option for the proposed algorithms.
  • Convergence behaviour: Algorithm 1 converges in fewer iterations, whereas Algorithm 2 converges faster in CPU time because it has closed-form updates.Both algorithms generate the same optimal objective values for IRS and NIRS cases.
  • IRS performance: The IRS structure enhances sum rate without additional transmit power compared with the system without IRS.The 2-bit phase shifter incurs only a small performance loss relative to continuous phase shifting.
  • IRS versus relay: At high transmit power, the IRS-aided system has higher energy efficiency than the relay-aided system, although the relay provides higher sum rate.The relay’s higher circuit consumption contributes to this energy-efficiency difference.
  • System scaling: Increasing IRS reflection elements provides larger spectral-efficiency gains than increasing BS transmit antennas, while passive IRS deployment is more energy-efficient.IRS gains over the no-IRS scheme remain significant even when M is as small as 4.
  • Multicast scaling: The sum rate decreases as users per multicast group increase because each group rate is limited by its worst-channel user.For |K_g| = 3, the sum rate increases slowly and tends to stabilize as the number of multicast groups grows.

APPENDIX A THE PROOF OF THEOREM 1

The appendix establishes convergence and stationarity properties for Algorithm 1 by transforming rate expressions, constructing lower-bound surrogates, and verifying KKT conditions.

  • Equivalent transformation: The rate expression is transformed into a jointly convex representation in auxiliary variables, enabling a first-order lower-bound surrogate.The variables are defined through t_k = e^H H_k f_g and r_k = r_{k,-g} + |t_k|^2.
  • Equivalent transformation: The transformed surrogate is substituted into the rate expression to complete the lower-bound construction.
  • Convergence: The objective sequence generated by Algorithm 1 is monotonic, and the iterates converge to a stable point because the feasible sets are bounded.The appendix then uses KKT conditions to characterize the converged solution.
  • KKT verification: The proof verifies complementary-slackness and stationarity relations for the precoding and reflection-coefficient subproblems.These conditions connect the surrogate subproblem solutions to the original optimization problem.
  • KKT verification: The max-min formulation is rewritten using auxiliary group-rate variables, and the resulting equations constitute the KKT conditions.

APPENDIX C THE PROOF OF THEOREM 3

The proof constructs quadratic minorizing surrogates for the precoder-related objective using directional-derivative matching and curvature bounds.

  • Precoder surrogate: A quadratic surrogate is proposed because the group objective f_g(F) is twice differentiable and concave.
  • Precoder surrogate: The surrogate is required to match the original objective’s directional derivative at the current iterate.This establishes the relevant first-order tightness condition.
  • Curvature condition: A sufficient lower-bound condition is obtained by comparing second derivatives along feasible line segments.The comparison is imposed for all γ ∈ [0, 1] and feasible precoders.
  • Curvature condition: The curvature expression is converted into quadratic form using vectorization and trace identities.
  • Curvature bound: Choosing M_g = α_g I with α_g = λ_min(Φ_g) provides the matrix used in the quadratic minorizer.The complex matrix Φ_g is bounded using the stated matrix inequalities.
  • Curvature bound: The lower bound on Φ_g is derived from eigenvalue and positive-semidefinite matrix inequalities, completing the surrogate construction.

APPENDIX D THE PROOF OF THEOREM 4

The proof constructs a quadratic minorizer for the reflection-coefficient subproblem by enforcing derivative matching and a second-order curvature inequality under unit-modulus constraints.

  • Reflection surrogate: Because f_g(e) is twice differentiable and concave, it is minorized at e_n by a quadratic function.
  • Reflection surrogate: The surrogate matches the directional derivative of f_g(e) at the current reflection vector.
  • Curvature condition: A sufficient lower-bound condition compares second-order derivatives along feasible line segments between e_n and e.The inequality is required over γ ∈ [0, 1] and feasible reflection vectors.
  • Curvature bound: The matrix N_g is selected to satisfy the surrogate conditions, with β_g chosen as λ_min(Ψ_g).The unit-modulus constraint gives e^H e = e_n^H e_n = M + 1.

APPENDIX E THE PROOF OF THEOREM 5

The appendix proves that the converged solution of Problem (24) satisfies its KKT conditions by combining optimality conditions for the precoding and reflection variables.

  • The converged solution {Fo, eo} is shown to satisfy the KKT conditions of Problem (24).
  • Because Fo is globally optimal for Problem (32), its Lagrangian KKT conditions, including the optimal multiplier τo, are established.
  • Substitution into the preceding conditions yields equations (89), (87), (90), and (91), which constitute the KKT conditions of Problem (24).
  • Because eo is locally optimal for Problem (45), the corresponding KKT conditions and optimal Lagrange multipliers are obtained.
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