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An Efficient Convex-Hull Relaxation Based Algorithm for Multi-User Discrete Passive Beamforming

Wenhai Lai, Zheyu Wu, Yi Feng, Kaiming Shen, Ya-Feng Liu

TL;DR

This work tackles max-min SINR optimization for an IRS-assisted downlink under discrete, $K$-ary phase shifts by introducing a convex-hull relaxation that replaces the discrete set $\mathcal X$ with conv$(\mathcal X)$, yielding a continuous reformulation that remains equivalent to the original problem when augmented with a sparsity penalty $\lambda \|\bm x\|_1$. An alternating projection/proximal gradient algorithm solves the reformulation efficiently, using a simplex projection for an auxiliary variable $\bm y$ on the simplex and a proximal step for $\bm x$ within $\mathrm{conv}(\mathcal X)$, achieving a per-iteration complexity of $\mathcal{O}(N^2 U^2)$. The paper proves that a finite $\lambda$ suffices to push solutions to the discrete vertices, ensuring equivalence to the original problem, and demonstrates strong empirical gains over state-of-the-art methods in simulations with up to $N=500$ IRS elements and $U$ users. The approach scales to active IRS and weighted sum-rate variants, offering a practical, efficient route to robust multi-user IRS design under discrete phase constraints. The method holds potential for real-world deployment by delivering improved fairness (min-SINR) and faster convergence in dense IRS-aided networks.

Abstract

Intelligent reflecting surface (IRS) is an emerging technology to enhance spatial multiplexing in wireless networks. This letter considers the discrete passive beamforming design for IRS in order to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among multiple users in an IRS-assisted downlink network. The main design difficulty lies in the discrete phase-shift constraint. Differing from most existing works, this letter advocates a convex-hull relaxation of the discrete constraints which leads to a continuous reformulated problem equivalent to the original discrete problem. This letter further proposes an efficient alternating projection/proximal gradient descent and ascent algorithm for solving the reformulated problem. Simulation results show that the proposed algorithm outperforms the state-of-the-art methods significantly.

An Efficient Convex-Hull Relaxation Based Algorithm for Multi-User Discrete Passive Beamforming

TL;DR

This work tackles max-min SINR optimization for an IRS-assisted downlink under discrete, -ary phase shifts by introducing a convex-hull relaxation that replaces the discrete set with conv, yielding a continuous reformulation that remains equivalent to the original problem when augmented with a sparsity penalty . An alternating projection/proximal gradient algorithm solves the reformulation efficiently, using a simplex projection for an auxiliary variable on the simplex and a proximal step for within , achieving a per-iteration complexity of . The paper proves that a finite suffices to push solutions to the discrete vertices, ensuring equivalence to the original problem, and demonstrates strong empirical gains over state-of-the-art methods in simulations with up to IRS elements and users. The approach scales to active IRS and weighted sum-rate variants, offering a practical, efficient route to robust multi-user IRS design under discrete phase constraints. The method holds potential for real-world deployment by delivering improved fairness (min-SINR) and faster convergence in dense IRS-aided networks.

Abstract

Intelligent reflecting surface (IRS) is an emerging technology to enhance spatial multiplexing in wireless networks. This letter considers the discrete passive beamforming design for IRS in order to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among multiple users in an IRS-assisted downlink network. The main design difficulty lies in the discrete phase-shift constraint. Differing from most existing works, this letter advocates a convex-hull relaxation of the discrete constraints which leads to a continuous reformulated problem equivalent to the original discrete problem. This letter further proposes an efficient alternating projection/proximal gradient descent and ascent algorithm for solving the reformulated problem. Simulation results show that the proposed algorithm outperforms the state-of-the-art methods significantly.
Paper Structure (6 sections, 2 theorems, 22 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 2 theorems, 22 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Convex-Hull Relaxation
  4. Proposed Algorithm
  5. Simulation Results
  6. Conclusion

Key Result

Lemma 1

All locally and globally optimal solutions of problem penalty_problem lie on the boundary of $\mathrm{conv}(\mathcal{X})$, which is denoted as $\partial\mathrm{conv}(\mathcal{X})$, so long as the parameter where $L_u = \frac{2\sqrt{N+1}\|\bm C_{uu}\|_F}{\sigma_u^2}\left(1+\frac{(N+1)\|\sum_{u^\prime \neq u}\bm C_{uu^\prime}\|_F}{\sigma_u^2}\right)$.

Figures (6)

  • Figure 1: Conventional relaxation $\hat{\mathcal{X}}$ vs. convex-hull $\mathrm{conv}(\mathcal{X})$ when $K=4$.
  • Figure 2: Two key steps in proving Lemma \ref{['coro:boundary']} and Theorem \ref{['prop:global_optimal']}. Panel (a) shows the definition of $rx_{n'}$. Panel (b) illustrates the computation of $|x_{n'}|$.
  • Figure 3: The IRS-assisted downlink network in our simulations.
  • Figure 4: Minimum SINR vs. $U$ when $N=500$.
  • Figure 5: Minimum SINR vs. $N$ when $U=9$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1