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A Riemannian Manifold Approach to Constrained Resource Allocation in ISAC

Shayan Zargari, Diluka Galappaththige, Chintha Tellambura, Vincent Poor

TL;DR

This work tackles constrained resource allocation for ISAC by formulating a non-convex joint beamforming problem that maximizes downlink sum rate while enforcing sensing beampattern and power constraints. It introduces Augmented Lagrangian Manifold Optimization (ALMO), combining Riemannian manifold optimization with an augmented Lagrangian to manage constraints and exploit manifold geometry. The method reformulates the problem with fractional programming, optimizes on a complex sphere manifold, and iteratively updates both beamformers and Lagrange multipliers, with convergence to stationary points guaranteed under standard conditions. Numerical results show that the proposed Iterative Manifold-Based Optimization (IMBO) yields substantial gains over SDR/SCA-based baselines (e.g., at $M=12$ antennas and $P_{ m max}=30$ dBm, around a $10.1 ext{ extdegree}$.% in sum rate improvement) and demonstrates fast convergence and favorable scalability, indicating practical viability for real-time ISAC resource management.

Abstract

This paper introduces a new resource allocation framework for integrated sensing and communication (ISAC) systems, which are expected to be fundamental aspects of sixth-generation networks. In particular, we develop an augmented Lagrangian manifold optimization (ALMO) framework designed to maximize communication sum rate while satisfying sensing beampattern gain targets and base station (BS) transmit power limits. ALMO applies the principles of Riemannian manifold optimization (MO) to navigate the complex, non-convex landscape of the resource allocation problem. It efficiently leverages the augmented Lagrangian method to ensure adherence to constraints. We present comprehensive numerical results to validate our framework, which illustrates the ALMO method's superior capability to enhance the dual functionalities of communication and sensing in ISAC systems. For instance, with 12 antennas and 30 dBm BS transmit power, our proposed ALMO algorithm delivers a 10.1% sum rate gain over a benchmark optimization-based algorithm. This work demonstrates significant improvements in system performance and contributes a new algorithmic perspective to ISAC resource management.

A Riemannian Manifold Approach to Constrained Resource Allocation in ISAC

TL;DR

This work tackles constrained resource allocation for ISAC by formulating a non-convex joint beamforming problem that maximizes downlink sum rate while enforcing sensing beampattern and power constraints. It introduces Augmented Lagrangian Manifold Optimization (ALMO), combining Riemannian manifold optimization with an augmented Lagrangian to manage constraints and exploit manifold geometry. The method reformulates the problem with fractional programming, optimizes on a complex sphere manifold, and iteratively updates both beamformers and Lagrange multipliers, with convergence to stationary points guaranteed under standard conditions. Numerical results show that the proposed Iterative Manifold-Based Optimization (IMBO) yields substantial gains over SDR/SCA-based baselines (e.g., at antennas and dBm, around a .% in sum rate improvement) and demonstrates fast convergence and favorable scalability, indicating practical viability for real-time ISAC resource management.

Abstract

This paper introduces a new resource allocation framework for integrated sensing and communication (ISAC) systems, which are expected to be fundamental aspects of sixth-generation networks. In particular, we develop an augmented Lagrangian manifold optimization (ALMO) framework designed to maximize communication sum rate while satisfying sensing beampattern gain targets and base station (BS) transmit power limits. ALMO applies the principles of Riemannian manifold optimization (MO) to navigate the complex, non-convex landscape of the resource allocation problem. It efficiently leverages the augmented Lagrangian method to ensure adherence to constraints. We present comprehensive numerical results to validate our framework, which illustrates the ALMO method's superior capability to enhance the dual functionalities of communication and sensing in ISAC systems. For instance, with 12 antennas and 30 dBm BS transmit power, our proposed ALMO algorithm delivers a 10.1% sum rate gain over a benchmark optimization-based algorithm. This work demonstrates significant improvements in system performance and contributes a new algorithmic perspective to ISAC resource management.
Paper Structure (27 sections, 2 theorems, 25 equations, 11 figures, 2 tables, 3 algorithms)

This paper contains 27 sections, 2 theorems, 25 equations, 11 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Previous Approaches for Beamforming Design
  3. Riemannian Manifold Optimization
  4. Motivation and Our Contribution
  5. System, Channel, and Signal Models
  6. System and Channel Models
  7. Transmission Model
  8. Communication and Sensing Performance
  9. Communication Performance
  10. Sensing Performance
  11. Problem Formulation
  12. Proposed Solution
  13. Optimization of $\boldsymbol{\mu}$ with Fixed $\mathbf{W}$
  14. Optimization on a Manifold
  15. Manifold optimization
  16. ...and 12 more sections

Key Result

Proposition 1

Let the ALMO algorithm be run with $\epsilon_{\min} = 0$, generating an infinite sequence $\{\epsilon_t\}$ converging to zero. At each iteration $t$, let Algorithm alg:CG identify a candidate solution $\mathbf{\tilde{W}}_{t+1}$ that meets the following condition: where $\mathbf{\tilde{W}}_{t+1}$ denotes a feasible global minimizer of $\text{(P5)}$. If there exists a limit point $\mathbf{\tilde{W}

Figures (11)

  • Figure 1: System model of an ISAC system: A $M$-antenna BS communicates with $K$ communication users and senses $N$ targets using a shared antenna array.
  • Figure 2: Key steps in MO.
  • Figure 3: Convergence rate under different system setups.
  • Figure 4: Sum rate versus penalty parameter $\rho$ (bottom x-axis) and constant $\theta_\epsilon$ (top x-axis) for the IMBO algorithm, differentiated by line style for each $\tau$ setting.
  • Figure 5: Average IMBO and CCPA running time versus number of users and BS antennas.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof