PaperlayerView exact revision on arXiv

Source-linked AI summary

IRS-Assisted Green Communication Systems: Provable Convergence and Robust Optimization

Xianghao Yu, Dongfang Xu, Derrick Wing Kwan Ng, Robert Schober

arXiv:2011.06484v1cs.ITeess.SPmath.IT

TL;DR

The paper addresses the lack of convergence-guaranteed power allocation for IRS-assisted multiuser MISO systems and the lack of comprehensive robust design under imperfect CSI. It jointly optimizes AP beamforming and IRS phases with convergent algorithms, and reports better performance than SDR-based AltMin alongside a speed–power trade-off between its perfect-CSI methods.

  • Problem

    Convergence-guaranteed power allocation and an accurate, comprehensive robust framework for IRS-assisted systems with imperfect CSI remain unavailable in prior work.

    Source: §I · p. 3

  • Method

    The paper jointly optimizes AP beamforming and IRS phase shifts using penalty-based AltMin and IA for perfect CSI, then extends AltMin to robust design with imperfect direct- and reflected-channel CSI.

    Source: §I · p. 4

  • Results

    The perfect-CSI algorithms outperform state-of-the-art SDR-based AltMin, while penalty-based AltMin converges faster at the cost of higher transmit power than IA.

    Source: §I · p. 4

  • Takeaways & Limitations

    A near-optimal pair of transmit-antenna and IRS-element counts can be found sequentially, avoiding the computationally expensive joint grid search.

    Source: F. System Power Consumption Minimization · p. 27

Abstract

from arXiv · show

Intelligent reflecting surfaces (IRSs) are regarded as a key enabler of green wireless communication, due to their capability of customizing favorable wireless propagation environments. In this paper, we investigate resource allocation for IRS-assisted green multiuser multiple-input single-output (MISO) systems. To minimize the total transmit power, both the beamforming vectors at the access point (AP) and the phase shifts at multiple IRSs are jointly optimized, while taking into account the minimum required quality-of-service (QoS) of multiple users. First, two novel algorithms, namely a penalty-based alternating minimization (AltMin) algorithm and an inner approximation (IA) algorithm, are developed to tackle the non-convexity of the formulated optimization problem when perfect channel state information (CSI) is available. Unlike existing designs that cannot ensure convergence, the proposed penalty-based AltMin and IA algorithms are guaranteed to converge to a stationary point and a Karush-Kuhn-Tucker (KKT) solution of the design problem, respectively. Second, the impact of imperfect knowledge of the CSI of the channels between the AP and the users is investigated. To this end, a non-convex robust optimization problem is formulated and the penalty-based AltMin algorithm is extended to obtain a stationary solution. Simulation results reveal a key trade-off between the speed of convergence and the achievable total transmit power for the two proposed algorithms. In addition, we show that the proposed algorithms can significantly reduce the total transmit power at the AP compared to various baseline schemes and that the optimal numbers of transmit antennas and IRS reflecting elements, which minimize the total power consumption of the considered system, are finite.

I. INTRODUCTION

The paper addresses green multiuser MISO design by jointly controlling AP beamforming and IRS reflections, while targeting gaps in convergence guarantees and robust CSI optimization.

I. INTRODUCTION

  • IRSs can customize wireless propagation with low-power passive elements, but their phase shifts must be designed jointly with AP beamforming to reduce power.The paper positions IRSs as a way to control propagation environments that conventional approaches treat as a “black box.”Source: §I · p. 2
  • Existing SDR-based AltMin designs lack convergence guarantees because randomized rank-one solutions may violate users’ QoS constraints.The paper identifies computationally efficient algorithms with convergence guarantees as an open design problem –.Source: §I · p. 3
  • For imperfect CSI, prior robust designs leave a gap: an accurate, comprehensive framework with provably convergent algorithms is not yet available.Prior work considered outage approximations or lower bounds on received signal power [19]–.Source: §I · p. 3
  • The proposed perfect-CSI algorithms outperform SDR-based AltMin, while penalty-based AltMin converges faster at the cost of higher transmit power than IA.For imperfect CSI, the penalty-based AltMin extension attains a stationary point of the robust problem.Source: §I · p. 4

II. SYSTEM MODEL AND EXISTING APPROACH

  • Section II introduces the system model and resource-allocation problem under perfect CSI, then discusses the existing approach and its limitations.Source: §II · p. 5

A. IRS-Assisted System Model and Problem Formulation

  • The design minimizes AP transmit power subject to each user’s minimum SINR, jointly choosing beamformers and IRS phases; unit-modulus constraints and coupled QoS constraints make it non-convex.The perfect-CSI formulation is presented as a theoretical performance upper bound, with CSI uncertainty considered later.Source: A. IRS-Assisted System Model and… · pp. 6–7

B. Existing Approach

The existing SDR-based AltMin approach alternates between beamforming and IRS phase-shift optimization, but its relaxation and randomization do not reliably preserve feasibility or convergence behavior.

B. Existing Approach

  • SDR-based AltMin decouples beamforming and phase-shift optimization, performing the two steps alternately.Beamformers are optimized conditioned on the current phase shifts, while the phase matrix is handled through a feasibility check.Source: B. Existing Approach · p. 7
  • Even optimal phase-step solutions cannot guarantee monotone objective values because phase updates change the beamforming feasible set, whose relation across iterations is difficult to quantify.The phase solutions may not be unique, so the successive feasible sets are difficult to compare.Source: B. Existing Approach · p. 7
  • The outage probability of QoS constraint C1 approaches one at a 2 dB target, so the normalized randomized vector almost never guarantees the required QoS.The outage probability is defined as the probability that users’ received SINRs fall below the target.Source: Figure 2 · p. 8, B. Existing Approach · p. 9
  • The SDR-based AltMin objective fluctuates significantly during iteration, with the convergence figure reporting the spread across realizations.The figure caption identifies the shaded area as plus or minus the standard deviation of convergence.Source: Figure 3 · p. 8, B. Existing Approach · pp. 8–9
  • Dropping the rank-one constraint in the SDR relaxation does not guarantee a rank-one solution, and Gaussian randomization does not ensure the QoS constraints are satisfied.The text identifies this as a feasibility limitation of the existing algorithm.Source: B. Existing Approach · pp. 8–9
  • Manually increasing the target SINR may help the randomized solution meet the original QoS constraint, but changes the feasible set and leaves optimality and monotonic convergence unguaranteed.The text presents this as a heuristic rather than a guarantee.Source: B. Existing Approach · p. 9

III. ALGORITHM DESIGN FOR IRS-ASSISTED MISO SYSTEMS WITH PERFECT CSI

The section identifies limitations of the existing SDR-based AltMin approach and introduces two algorithms with convergence guarantees for the reformulated problem.

A. Problem Reformulation

  • The reformulation expresses the numerator using an effective channel that includes both reflected and direct AP-to-user channels.Source: A. Problem Reformulation · pp. 9–10
  • The existing SDR-based AltMin approach lacks guaranteed objective monotonicity and may produce solutions that violate user QoS constraints.These limitations motivate the proposed algorithms for the reformulated problem.Source: A. Problem Reformulation · p. 10

III. ALGORITHM DESIGN FOR IRS-ASSISTED MISO SYSTEMS WITH PERFECT CSI

  • The proposed algorithms target the reformulated problem while addressing the existing approach’s feasibility-check and QoS-feasibility limitations.Source: A. Problem Reformulation · p. 10

B. Proposed Penalty-Based AltMin Algorithm

The penalty-based AltMin algorithm uses normalization and a penalty-based SCA step to avoid the existing feasibility check and converge to a stationary point.

B. Proposed Penalty-Based AltMin Algorithm

  • The beamforming subproblem is equivalent to the classical beamforming problem without IRSs and can be solved optimally.Source: B. Proposed Penalty-Based AltMin… · p. 11
  • The normalization makes transmit power an objective variable in both alternating subproblems, guaranteeing a monotonic objective sequence and avoiding the feasibility check.The normalization also helps avoid the feasibility check when optimizing the IRS reflecting elements.Source: B. Proposed Penalty-Based AltMin… · p. 11
  • A penalty-based SCA step handles the remaining rank-one constraint, with objective values forming a non-increasing sequence that converges to a stationary point.The penalized problem is treated as a difference-of-convex objective, and SCA uses a first-order Taylor approximation.Source: B. Proposed Penalty-Based AltMin… · p. 12
  • The overall two-block Gauss-Seidel AltMin algorithm is guaranteed to converge to a stationary point of problem (10) in polynomial time.Source: B. Proposed Penalty-Based AltMin… · pp. 12–13

C. Proposed Inner Approximation (IA) Algorithm

The IA algorithm jointly updates beamforming matrices and IRS variables using convex approximations, and converges to a KKT solution.

C. Proposed Inner Approximation (IA) Algorithm

IV. ALGORITHM DESIGN FOR IRS-ASSISTED MISO SYSTEMS WITH IMPERFECT CSI

The section introduces a CSI uncertainty model and robust resource allocation for IRS-assisted multiuser MISO systems.

IV. ALGORITHM DESIGN FOR IRS-ASSISTED MISO SYSTEMS WITH IMPERFECT CSI

  • The section introduces a CSI uncertainty model for IRS-assisted multiuser MISO systems.Source: §IV · p. 15
  • It formulates a robust resource allocation problem for the considered systems.Source: §IV · p. 15
  • It develops an effective algorithm to jointly optimize AP beamformers and IRS reflecting elements.Source: §IV · p. 15

A. Channel State Information (CSI)

Reflected-channel CSI is harder to estimate than direct-link CSI because passive IRSs cannot use active RF chains to receive and transmit estimation signals. The paper models uncertainty in both cascaded reflected and direct channels, while noting an empirical rank-one property of the IA solution.

A. Channel State Information (CSI)

  • The IA algorithm reportedly converges to a rank-one solution even when constraint C3 is dropped, although the paper does not prove this tightness.The proof is described as intricate and beyond the paper’s scope.Source: A. Channel State Information (CSI) · p. 15
  • Passive IRSs cannot receive and transmit signals for channel estimation, so estimation yields the cascaded reflected channel rather than the two individual reflected channels.Direct AP-to-user links can be estimated conventionally, whereas estimating the reflected channels is more challenging.Source: A. Channel State Information (CSI) · pp. 15–16
  • The CSI model uses norm-bounded errors for both cascaded reflected channels and direct links, with uncertainty levels tied to estimation and quantization accuracy.The model can capture noisy estimation, IRS phase-shift quantization errors, and limited feedback.Source: A. Channel State Information (CSI) · p. 16

B. Problem Formulation

The robust design jointly optimizes AP beamformers and IRS phase shifts under norm-bounded CSI uncertainty. The paper converts the resulting infinitely many robust QoS inequalities into finitely many LMIs and extends penalty-based AltMin to solve the problem.

B. Problem Formulation

  • The robust formulation jointly designs AP beamformers and IRS phase shifts to minimize transmit power under CSI uncertainty.The formulation uses norm-bounded uncertainty for the channels.Source: B. Problem Formulation · pp. 16, 18
  • Continuous uncertainty sets make constraint C5 an infinite family of non-convex inequalities, which the paper transforms into finitely many LMIs using the S-Procedure.The resulting formulation has K LMIs for constraint C5.Source: B. Problem Formulation · pp. 17–18
  • Because the robust LMI constraints make convex feasible-set approximations difficult, the paper extends penalty-based AltMin to the robust problem.The authors state that these constraints make the IA method difficult to apply to robust design.Source: B. Problem Formulation · p. 18
  • Unlike prior designs that typically model uncertainty only in reflected channels, this formulation accounts for CSI uncertainty in all communication links.The paper says prior safe approximations for direct-link uncertainty could restrict the feasible set and may increase total transmit power.Source: B. Problem Formulation · p. 18

C. Proposed Penalty-Based AltMin Algorithm

The penalty-based AltMin procedure alternates between optimizing the beamforming matrices and the IRS phase variables. Its beamforming relaxation has an optimal solution without the rank-one constraint, and the robust-case algorithm retains the earlier convergence and complexity analysis.

C. Proposed Penalty-Based AltMin Algorithm

  • For fixed IRS phases, the algorithm optimizes the beamforming matrices, and the relaxation has an optimal solution without the rank-one constraint.The theorem establishes this property for problem (37).Source: C. Proposed Penalty-Based AltMin… · p. 19
  • With beamforming matrices fixed, the next alternating step optimizes the IRS phase variable.The passage identifies this as the phase-variable subproblem in iteration t+1.Source: C. Proposed Penalty-Based AltMin… · p. 19
  • The robust-case SCA and penalty-based AltMin procedures replace the earlier subproblems with problems (38) and (37), while retaining the earlier convergence and complexity analysis.The passage says the analysis is identical to that at the end of Section III-B.Source: C. Proposed Penalty-Based AltMin… · p. 19

V. SIMULATION RESULTS

The simulations model an IRS-assisted multiuser MISO system with users distributed in a cell sector and IRSs at the cell edge. They compare the proposed algorithms with SDR-AltMin and two baseline schemes.

A. Simulation Setup

SIMULATION PARAMETERS

  • The channel setup uses Ricean fading for AP-to-IRS and IRS-to-user links, Rayleigh fading for direct links, and identical user SINR targets and noise powers.The direct links are modeled as non-LoS channels; simulation parameters are listed in Table I.Source: SIMULATION PARAMETERS · p. 20

B. Baseline Schemes

  • The benchmarks include SDR-AltMin, a conventional MISO system without IRSs, and an IRS scheme with random phase shifts; SDR-AltMin receives a feasibility step when needed.For fair comparison, its maximum iterations are matched to the proposed algorithms’ convergence iterations.Source: B. Baseline Schemes · pp. 20–21

C. Convergence of the Proposed Algorithms

The simulations show a trade-off between convergence speed and transmit-power performance, while examining power minimization under perfect and imperfect CSI. They also find a finite system-power-minimizing antenna and reflecting-element configuration, with a sequential search suggested for finding a near-optimal pair.

C. Convergence of the Proposed Algorithms

  • For Nt = 4 and M = 6, penalty-based AltMin and IA converge within 15 and 300 iterations on average; for Nt = 6 and M = 10, they take 30 and 800.IA achieves lower average transmit power, while penalty-based AltMin converges faster, reflecting a speed–power trade-off.Source: C. Convergence of the Proposed… · pp. 21–22

D. Average Transmit Power Versus the Minimum Required SINR

  • With perfect CSI, average AP transmit power rises with the minimum required SINR, while IRS deployment and the proposed algorithms reduce power relative to the stated baselines.The proposed algorithms outperform random phase shifts and SDR-based AltMin, which cannot guarantee convergence.Source: D. Average Transmit Power Versus the… · p. 22, Figure 7 · p. 23
  • With imperfect CSI, the proposed penalty-based AltMin reduces transmit power against three baselines; baseline 1 uses less power than baseline 2 because it faces uncertainty only on direct links.The performance gap between SDR-based AltMin and the proposed scheme is larger than in the perfect-CSI case; Gaussian randomization is less likely to yield a feasible solution under robust QoS constraints.Source: D. Average Transmit Power Versus the… · p. 23

E. CSI Uncertainty

  • As channel-estimation error variance κ increases, average transmit power rises, and the proposed penalty-based AltMin still outperforms the baselines, though its gain over baseline 1 decreases.The IRS-assisted schemes must account for uncertainty in both direct and reflected channels, whereas baseline 1 accounts only for direct-link uncertainty.Source: E. CSI Uncertainty · p. 24
  • At a 5.8 dB SINR threshold, outage probability is 100% for baseline schemes 1 and 2, 85% for SDR-based AltMin, and 20% for the proposed robust scheme.At thresholds no higher than the 5 dB minimum required SINR, the proposed scheme and three baselines have zero outage; the non-robust scheme has outage probability one at 5 dB.Source: E. CSI Uncertainty · p. 25

F. System Power Consumption Minimization

  • System power consumption includes transmit power and power consumed by the AP’s static circuitry, RF chains, and IRS controllers.The paper defines total system power as Ptotal = P/η + Ps + NtPRF + MPIRS.Source: F. System Power Consumption Minimization · p. 25
  • Nt = 6 and M = 12 minimize average system power consumption under CSI uncertainty; system power is more sensitive to Nt than M, enabling a near-optimal sequential search.System power first decreases and then increases as antenna and IRS-element counts grow: added degrees of freedom reduce transmit power, but component power eventually outweighs that reduction.Source: F. System Power Consumption Minimization · pp. 26–27

VI. CONCLUSIONS

  • The conclusions identify penalty-based AltMin as fast-converging to a stationary point and IA as reaching a locally optimal KKT solution under perfect CSI; AltMin is extended to a stationary solution under CSI uncertainty.The paper’s simulations also identify a small antenna count and relatively large IRS-element count as beneficial for minimizing total system power.Source: §VI · p. 27

APPENDIX

The appendix proofs relax rank-one constraints to obtain convex problems and use strong duality and KKT analysis. The second proof also establishes a rank-one solution for the beamforming matrix.

A. Proof of Theorem 1

  • For Theorem 1, relaxing rank-one constraint C4 yields a convex problem satisfying Slater’s constraint qualification, so strong duality holds; the proof then examines KKT conditions.Source: A. Proof of Theorem 1 · pp. 27–28

B. Proof of Theorem 2

  • For Theorem 2, relaxing C4 yields a convex problem satisfying Slater’s constraint qualification, and the proof shows Rank(Wk) ≤ 1 can always be obtained.Source: B. Proof of Theorem 2 · pp. 28–29

Related papers

Loading 2011.06484v1…