pith. sign in

arxiv: 2509.04873 · v2 · pith:FD2N5WFCnew · submitted 2025-09-05 · 📡 eess.SP

Movable IRS-Aided ISAC Systems: Joint Beamforming and Position Optimization

Yue Geng , Tee Hiang Cheng , Kai Zhong , Kah Chan Teh , Qingqing Wu This is my paper

Pith reviewed 2026-05-21 23:17 UTC · model grok-4.3

classification 📡 eess.SP
keywords movable IRSintegrated sensing and communicationposition optimizationbeamformingRiemannian manifold optimizationpower minimizationISAC systems
0
0 comments X

The pith

Movable IRS reduces required transmit power in ISAC by optimizing element positions jointly with beamforming.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines movable intelligent reflecting surfaces in integrated sensing and communication systems to cut the power needed while meeting quality-of-service targets for both functions. It jointly tunes MIRS element positions, reflection coefficients, transmit beamforming vectors, and receive filters. Two practical control modes are defined: element-wise adjustment of each reflector and array-wise movement of grouped elements. The joint non-convex problem is solved by mapping variables onto a product Riemannian manifold and applying a penalty-augmented RBFGS update that runs the blocks in parallel. Simulations indicate that both modes consume less power than a fixed-position IRS, with element-wise control reaching the lowest values and array-wise control offering faster computation.

Core claim

Allowing the positions of IRS reflecting elements to be adjusted, and jointly optimizing those positions together with reflection coefficients, transmit beamforming, and receive filters, lowers the total transmit power required to satisfy given sensing and communication QoS constraints compared with a conventional fixed IRS; the optimization is performed by constructing a product Riemannian manifold space and updating all variables in parallel through a penalty-based transformation combined with the RBFGS algorithm.

What carries the argument

Product Riemannian manifold optimization (PRMO) method, which builds a product manifold space and performs parallel variable updates via penalty transformation and the Riemannian Broyden-Fletcher-Goldfarb-Shanno algorithm.

If this is right

  • Element-wise position control yields the lowest achievable power across a range of system parameters.
  • Array-wise control delivers a slightly higher power but with markedly lower computational cost.
  • Both MIRS variants outperform fixed IRS in power minimization for the ISAC objective.
  • The two control granularities let designers trade performance against implementation complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If positions can be updated on the order of channel coherence time, MIRS could track moving users or changing scatterers without extra power margin.
  • The same manifold formulation might be reused for other movable-antenna ISAC setups that include user-side mobility.
  • Energy savings demonstrated here would compound in dense deployments where many surfaces share a common power budget.
  • Real-world validation would require measuring how quickly and accurately element positions can be physically adjusted.

Load-bearing premise

The non-convex joint optimization over MIRS positions, reflection coefficients, beamforming, and filters can be solved to a high-quality local optimum by the proposed product Riemannian manifold optimization method.

What would settle it

A set of Monte Carlo trials in which the MIRS schemes require equal or higher power than a fixed IRS to meet the same sensing and communication QoS targets would falsify the performance advantage.

Figures

Figures reproduced from arXiv: 2509.04873 by Kah Chan Teh, Kai Zhong, Qingqing Wu, Tee Hiang Cheng, Yue Geng.

Figure 1
Figure 1. Figure 1: The proposed MIRS-aided ISAC system. are summarized as follows: • We investigate the power minimization problem for MIRS-aided ISAC systems, aiming to minimize the power required for satisfying the communication rate and sensing signal-to-interference-plus-noise-ratio (SINR) thresholds for multiple communication users (CUs) and targets. For the joint beamforming and position optimization with both element-… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of the ME arrays with (a) a = 2 and (b) a = 3. B. Channel Modeling for MIRS With Array-Wise Control We then introduce the channel modeling when array-wise control is implemented for the MIRS. For convenience, we consider square arrays composed of a perfect square number of elements, with each row (or column) containing a elements, resulting in a total of a 2 elements in each array. The spacing bet… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of Algorithm 3 with different numbers of CU and target [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of Algorithm 3 under element-wise control and array [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Transmit power versus number of IRS elements. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Transmit power versus normalized IRS region size [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

Driven by intelligent reflecting surface (IRS) and movable antenna (MA) technologies, movable IRS (MIRS) has been proposed to improve the adaptability and performance of conventional IRS, enabling flexible adjustment of the IRS reflecting element positions. This paper investigates MIRS-aided integrated sensing and communication (ISAC) systems. The objective is to minimize the power required for satisfying the quality-of-service (QoS) of sensing and communication by jointly optimizing the MIRS element positions, IRS reflection coefficients, transmit beamforming, and receive filters. To balance the performance-cost trade-off, we proposed two MIRS schemes: element-wise control and array-wise control, where the positions of individual reflecting elements and arrays consisting of multiple elements are controllable, respectively. To address the joint beamforming and position optimization, a product Riemannian manifold optimization (PRMO) method is proposed, where the variables are updated over a constructed product Riemannian manifold space (PRMS) in parallel via penalty-based transformation and Riemannian Broyden-Fletcher-Goldfarb-Shanno (RBFGS) algorithm. Simulation results demonstrate that the proposed MIRS outperforms conventional IRS in power minimization with both element-wise control and array-wise control. Specifically, with different system parameters, the minimum power is achieved by the MIRS with the element-wise control scheme, while suboptimal solution and higher computational efficiency are achieved by the MIRS with array-wise control scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies movable IRS (MIRS) in ISAC systems and proposes two position-control schemes (element-wise and array-wise). It formulates a power-minimization problem subject to sensing and communication QoS constraints, jointly optimizing MIRS element/array positions, reflection coefficients, transmit beamforming, and receive filters. The non-convex problem is addressed via a product Riemannian manifold optimization (PRMO) method that employs a penalty transformation and the RBFGS algorithm on a constructed product manifold. Simulation results are presented to show that both MIRS schemes achieve lower minimum power than a conventional fixed-position IRS, with element-wise control yielding the best performance and array-wise control offering a complexity-performance trade-off.

Significance. If the PRMO solver reliably attains high-quality stationary points, the work would provide concrete evidence that position mobility in IRS can reduce the power needed to meet joint ISAC QoS targets relative to fixed IRS. The approach of embedding position variables directly into a Riemannian product manifold and applying established RBFGS updates is a technically coherent way to handle the coupled continuous constraints; this is a positive methodological contribution that could be reused in other movable-antenna or fluid-antenna ISAC settings.

major comments (2)
  1. [§III] §III (PRMO algorithm description): The manuscript provides no convergence analysis, no iteration-count or objective-value plots, and no multi-start or multi-initialization statistics for the penalty-augmented RBFGS procedure. Because the headline claim (MIRS power savings versus fixed IRS) is obtained exclusively from the solutions returned by this solver, the absence of such validation leaves open the possibility that reported gains partly reflect more thorough exploration of the movable-position search space rather than intrinsic superiority of MIRS.
  2. [§IV] §IV (Simulation results): The abstract and results section do not report the number of Monte-Carlo channel realizations, the precise channel model parameters (path-loss exponents, Rician factors, etc.), or any statistical significance test on the observed power reductions. These omissions make it difficult to assess whether the claimed outperformance of element-wise and array-wise MIRS over fixed IRS is robust across typical ISAC operating regimes.
minor comments (2)
  1. [§II] Notation for the product manifold and the penalty parameter schedule should be introduced earlier and used consistently; several symbols (e.g., the manifold retraction operator) appear without prior definition.
  2. [§IV] Figure captions for the power-versus-iteration and power-versus-element-count plots should explicitly state the number of random initializations used and whether the curves represent the best, median, or average outcome.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [§III] §III (PRMO algorithm description): The manuscript provides no convergence analysis, no iteration-count or objective-value plots, and no multi-start or multi-initialization statistics for the penalty-augmented RBFGS procedure. Because the headline claim (MIRS power savings versus fixed IRS) is obtained exclusively from the solutions returned by this solver, the absence of such validation leaves open the possibility that reported gains partly reflect more thorough exploration of the movable-position search space rather than intrinsic superiority of MIRS.

    Authors: We agree that additional validation of the PRMO solver strengthens the claims. In the revised manuscript we will add a dedicated subsection in Section III that presents a convergence analysis of the penalty-augmented RBFGS procedure on the product manifold, including a proof sketch for stationarity under the penalty transformation. We will also include new figures that plot the objective value and constraint violation versus iteration count for representative channel realizations, together with a table summarizing results from 20 independent random initializations (reporting mean, standard deviation, and best/worst objective values). These additions will demonstrate that the reported power savings are reproducible and attributable to the movable IRS architecture rather than initialization bias. revision: yes

  2. Referee: [§IV] §IV (Simulation results): The abstract and results section do not report the number of Monte-Carlo channel realizations, the precise channel model parameters (path-loss exponents, Rician factors, etc.), or any statistical significance test on the observed power reductions. These omissions make it difficult to assess whether the claimed outperformance of element-wise and array-wise MIRS over fixed IRS is robust across typical ISAC operating regimes.

    Authors: We acknowledge these reporting omissions. The revised manuscript will explicitly state that all numerical results are averaged over 1000 independent Monte-Carlo channel realizations. We will add a table (or expanded paragraph in Section IV) listing all channel parameters, including path-loss exponents, Rician K-factors, noise variances, and carrier frequency. In addition, the performance figures will be updated to display error bars representing one standard deviation across the realizations. While we did not conduct formal hypothesis testing, the consistent ordering of the three schemes across multiple system parameters (number of elements, SNR targets, and array sizes) already indicates robustness; the added statistics will make this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity in optimization formulation or performance claims

full rationale

The paper formulates a standard non-convex joint optimization problem over MIRS positions, reflection coefficients, transmit beamforming and receive filters for an ISAC system, then applies an established product Riemannian manifold optimization procedure (penalty transformation plus RBFGS) to obtain numerical solutions. Simulation results comparing element-wise and array-wise MIRS control against fixed IRS are direct outputs of this solver rather than quantities defined in terms of themselves or fitted parameters renamed as predictions. No self-definitional steps, no load-bearing self-citations that reduce the central claim to prior author work, and no ansatz smuggled via citation appear in the provided text. The derivation chain remains self-contained as a conventional numerical optimization pipeline whose outputs are independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract introduces no explicit free parameters or invented physical entities; the approach relies on standard wireless channel assumptions and manifold optimization techniques.

axioms (1)
  • domain assumption Wireless channel models and sensing/communication QoS metrics accurately represent the physical environment for the purpose of power minimization.
    Standard modeling premise invoked when claiming simulation-based performance gains in ISAC papers.

pith-pipeline@v0.9.0 · 5789 in / 1200 out tokens · 57554 ms · 2026-05-21T23:17:37.590361+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    The integrated sensing and communication revolution for 6G: Vision, techniques, and applications,

    N. Gonz ´alez-Prelcicet al., “The integrated sensing and communication revolution for 6G: Vision, techniques, and applications,”Proc. IEEE, vol. 112, no. 7, pp. 676–723, 2024

    work page 2024

  2. [2]

    Enabling joint communication and radar sensing in mobile networks—A survey,

    J. A. Zhanget al., “Enabling joint communication and radar sensing in mobile networks—A survey,”IEEE Commun. Surv. Tutor ., vol. 24, no. 1, pp. 306–345, 2022

    work page 2022

  3. [3]

    Integrated sensing and communications: Toward dual- functional wireless networks for 6G and beyond,

    F. Liuet al., “Integrated sensing and communications: Toward dual- functional wireless networks for 6G and beyond,”IEEE J. Sel. Areas Commun., vol. 40, no. 6, pp. 1728–1767, 2022

    work page 2022

  4. [4]

    Integrated sensing and communications: Recent advances and ten open challenges,

    S. Luet al., “Integrated sensing and communications: Recent advances and ten open challenges,”IEEE Internet Things J., vol. 11, no. 11, pp. 19 094–19 120, 2024

    work page 2024

  5. [5]

    Movable antennas for wireless commu- nication: Opportunities and challenges,

    L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless commu- nication: Opportunities and challenges,”IEEE Commun. Mag., vol. 62, no. 6, pp. 114–120, 2024

    work page 2024

  6. [6]

    Flexible-position MIMO for wireless communications: Fundamentals, challenges, and future directions,

    J. Zhenget al., “Flexible-position MIMO for wireless communications: Fundamentals, challenges, and future directions,”IEEE Wirel. Commun., vol. 31, no. 5, pp. 18–26, 2024

    work page 2024

  7. [7]

    Modeling and performance analysis for movable antenna enabled wireless communications,

    L. Zhu, W. Ma, and R. Zhang, “Modeling and performance analysis for movable antenna enabled wireless communications,”IEEE Trans. Wirel. Commun., vol. 23, no. 6, pp. 6234–6250, 2024

    work page 2024

  8. [8]

    Multi-beam forming with movable- antenna array,

    W. Ma, L. Zhu, and R. Zhang, “Multi-beam forming with movable- antenna array,”IEEE Commun. Lett., vol. 28, no. 3, pp. 697–701, 2024

    work page 2024

  9. [9]

    Movable-antenna enhanced multiuser communication via antenna position optimization,

    L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,”IEEE Trans. Wirel. Commun., vol. 23, no. 7, pp. 7214–7229, 2024

    work page 2024

  10. [10]

    Weighted sum-rate maxi- mization for movable antenna-enhanced wireless networks,

    B. Feng, Y . Wu, X.-G. Xia, and C. Xiao, “Weighted sum-rate maxi- mization for movable antenna-enhanced wireless networks,”IEEE Wirel. Commun. Lett., vol. 13, no. 6, pp. 1770–1774, 2024

    work page 2024

  11. [11]

    MIMO capacity characterization for movable antenna systems,

    W. MA, L. Zhu and R. Zhang, “MIMO capacity characterization for movable antenna systems,”IEEE Trans. Wirel. Commun., vol. 23, no. 4, pp. 3392–3407, 2024

    work page 2024

  12. [12]

    Exploiting six-dimensional mov- able antenna for wireless sensing,

    X. Shao, R. Zhang, and R. Schober, “Exploiting six-dimensional mov- able antenna for wireless sensing,”IEEE Wirel. Commun. Lett., vol. 14, no. 2, pp. 265–269, 2025

    work page 2025

  13. [13]

    Movable antenna enhanced wireless sensing via antenna position optimization,

    W. Ma, L. Zhu, and R. Zhang, “Movable antenna enhanced wireless sensing via antenna position optimization,”IEEE Trans. Wirel. Com- mun., vol. 23, no. 11, pp. 16 575–16 589, 2024

    work page 2024

  14. [14]

    Movable antenna enabled integrated sensing and communication,

    W. Lyu, S. Yang, Y . Xiu, Z. Zhang, C. Assi, and C. Yuen, “Movable antenna enabled integrated sensing and communication,”IEEE Trans. Wirel. Commun., vol. 24, no. 4, pp. 2862–2875, 2025

    work page 2025

  15. [15]

    Movable antenna- aided near-field integrated sensing and communication,

    J. Ding, Z. Zhou, X. Shao, B. Jiao, and R. Zhang, “Movable antenna- aided near-field integrated sensing and communication,”IEEE Trans. Wirel. Commun., Early Access, 2025

    work page 2025

  16. [16]

    A survey on integrated sensing and communication with intelligent metasurfaces: Trends, challenges, and opportunities,

    A. Magbool, V . Kumar, Q. Wu, M. D. Renzo, and M. F. Flanagan, “A survey on integrated sensing and communication with intelligent metasurfaces: Trends, challenges, and opportunities,”IEEE Open J. Commun. Soc., Early Access, 2025

    work page 2025

  17. [17]

    Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,

    Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”IEEE Commun. Mag., vol. 58, no. 1, pp. 106–112, 2020

    work page 2020

  18. [18]

    An overview of signal processing techniques for RIS/IRS- aided wireless systems,

    C. Panet al., “An overview of signal processing techniques for RIS/IRS- aided wireless systems,”IEEE J. Sel. Top. Signal Process., vol. 16, no. 5, pp. 883–917, 2022

    work page 2022

  19. [19]

    Integrated sensing and communication in IRS-assisted high-mobility systems: De- sign, analysis, and optimization,

    X. Peng, Q. Tao, X. Hu, R. Jin, C. Huang, and X. Chen, “Integrated sensing and communication in IRS-assisted high-mobility systems: De- sign, analysis, and optimization,”IEEE Trans. Wirel. Commun., vol. 23, no. 11, pp. 16 107–16 122, 2024

    work page 2024

  20. [20]

    IRS-based integrated location sensing and communication for mmWave SIMO systems,

    X. Hu, C. Liu, M. Peng, and C. Zhong, “IRS-based integrated location sensing and communication for mmWave SIMO systems,”IEEE Trans. Wirel. Commun., vol. 22, no. 6, pp. 4132–4145, 2023

    work page 2023

  21. [21]

    Joint beamforming for CRB-constrained IRS-aided ISAC system via product manifold methods,

    Y . Geng, T. Hiang Cheng, K. Zhong, K. Chan Teh, and Q. Wu, “Joint beamforming for CRB-constrained IRS-aided ISAC system via product manifold methods,”IEEE Trans. Wirel. Commun., vol. 24, no. 1, pp. 691–705, 2025

    work page 2025

  22. [22]

    Cram ´er-Rao bound minimization for IRS-enabled multiuser integrated sensing and communications,

    X. Song, X. Qin, J. Xu, and R. Zhang, “Cram ´er-Rao bound minimization for IRS-enabled multiuser integrated sensing and communications,” IEEE Trans. Wirel. Commun., vol. 23, no. 8, pp. 9714–9729, 2024

    work page 2024

  23. [23]

    SNR/CRB-constrained joint beamforming and reflection designs for RIS-ISAC systems,

    R. Liu, M. Li, Q. Liu, and A. Lee Swindlehurst, “SNR/CRB-constrained joint beamforming and reflection designs for RIS-ISAC systems,”IEEE Trans. Wirel. Commun., vol. 23, no. 7, pp. 7456–7470, 2024

    work page 2024

  24. [24]

    Integrating movable antennas and intelligent reflecting surfaces (MA-IRS): Fundamentals, practical solutions, and opportuni- ties,

    Q. Wuet al., “Integrating movable antennas and intelligent reflecting surfaces (MA-IRS): Fundamentals, practical solutions, and opportuni- ties,”arXiv preprint arXiv:2506.14636, 2025

    work page arXiv 2025

  25. [25]

    Movable antennas meet intelligent reflecting surface: When do we need movable antennas?

    X. Wei, W. Mei, Q. Wu, B. Ning, and Z. Chen, “Movable antennas meet intelligent reflecting surface: When do we need movable antennas?” in WCNC, 2025

    work page 2025

  26. [26]

    Sum- rate optimization for RIS-aided multiuser communications with movable antenna,

    Y . Sun, H. Xu, B. Ning, Z. Cheng, C. Ouyang, and H. Yang, “Sum- rate optimization for RIS-aided multiuser communications with movable antenna,”IEEE Wirel. Commun. Lett., vol. 14, no. 2, pp. 450–454, 2025

    work page 2025

  27. [27]

    Movable antenna-enabled RIS-Aided integrated sensing and communication,

    H. Wu, H. Ren, C. Pan, and Y . Zhang, “Movable antenna-enabled RIS-Aided integrated sensing and communication,”IEEE Trans. Cogn. Commun. Netw., Early Access, 2025

    work page 2025

  28. [28]

    Movable-antenna aided se- cure transmission for RIS-ISAC systems,

    Y . Ma, K. Liu, Y . Liu, L. Zhu, and Z. Xiao, “Movable-antenna aided se- cure transmission for RIS-ISAC systems,”IEEE Trans. Wirel. Commun., Early Access, 2025

    work page 2025

  29. [29]

    Movable intelligent surface (MIS) for wireless communications: Architecture, modeling, algorithm, and prototyping,

    Z. Zheng, Q. Wu, W. Chen, X. Wu, and W. Zhu, “Movable intelligent surface (MIS) for wireless communications: Architecture, modeling, algorithm, and prototyping,”arXiv preprint arXiv:2412.19071, 2024

    work page arXiv 2024

  30. [30]

    Intelligent reflecting surface-aided wireless communication with movable elements,

    G. Hu et al., “Intelligent reflecting surface-aided wireless communication with movable elements,”IEEE Wirel. Commun. Lett., vol. 13, no. 4, pp. 1173–1177, 2024

    work page 2024

  31. [31]

    Joint beamforming and antenna position optimization for IRS-aided multi-user movable antenna systems,

    Y . Geng, T. H. Cheng, K. Zhong, K. C. Teh, and Q. Wu, “Joint beamforming and antenna position optimization for IRS-aided multi-user movable antenna systems,”arXiv preprint arXiv:2410.00634, 2025

    work page arXiv 2025

  32. [32]

    Exploiting movable elements of intelligent reflecting surface for enhancement of integrated sensing and communication,

    X. Peng, Q. Tao, Y . L. Guan, and X. Chen, “Exploiting movable elements of intelligent reflecting surface for enhancement of integrated sensing and communication,”IEEE Trans. Wirel. Commun., Early Access, 2025

    work page 2025

  33. [33]

    Channel estimation for movable antenna communication systems: A framework based on compressed sensing,

    Z. Xiao et al., “Channel estimation for movable antenna communication systems: A framework based on compressed sensing,”IEEE Trans. Wirel. Commun., vol. 23, no. 9, pp. 11 814–11 830, 2024

    work page 2024

  34. [34]

    Boumal,An Introduction to Optimization on Smooth Manifolds

    N. Boumal,An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, 2023

    work page 2023

  35. [35]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical Optimization. Springer New York, NY , 2006

    work page 2006

  36. [36]

    Steering exact penalty meth- ods for nonlinear programming,

    R. H. Byrd, J. Nocedal, and R. A. Waltz, “Steering exact penalty meth- ods for nonlinear programming,”Optimization Methods and Software, vol. 23, no. 2, pp. 197–213, 2008

    work page 2008

  37. [37]

    Simple algorithms for optimization on Rieman- nian manifolds with constraints,

    C. Liu and N. Boumal, “Simple algorithms for optimization on Rieman- nian manifolds with constraints,”Appl. Math. Optim., vol. 82, no. 3, pp. 949–981, 2020

    work page 2020

  38. [38]

    On smoothing exact penalty functions for convex constrained optimization,

    M. c. Pinar and S. A. Zenios, “On smoothing exact penalty functions for convex constrained optimization,”SIAM J. Optim., vol. 4, no. 3, pp. 486–511, 1994

    work page 1994

  39. [39]

    A. P. Ruszczy ´nski,Nonlinear Optimization. Princeton university press, 2006, vol. 13

    work page 2006

  40. [40]

    A Riemannian BFGS method for nonconvex optimization problems,

    W. Huang, P.-A. Absil, and K. A. Gallivan, “A Riemannian BFGS method for nonconvex optimization problems,” inENUMATH. Springer, 2016, pp. 627–634

    work page 2016

  41. [41]

    Adaptive, limited-memory BFGS algo- rithms for unconstrained optimization,

    P. T. Boggs and R. H. Byrd, “Adaptive, limited-memory BFGS algo- rithms for unconstrained optimization,”SIAM J. Optim., vol. 29, no. 2, pp. 1282–1299, 2019

    work page 2019

  42. [42]

    A Riemannian limited-memory BFGS algorithm for computing the matrix geometric mean,

    X. Yuan, W. Huang, P.-A. Absil, and K. A. Gallivan, “A Riemannian limited-memory BFGS algorithm for computing the matrix geometric mean,”Procedia Computer Science, vol. 80, pp. 2147–2157, 2016

    work page 2016

  43. [43]

    Global rates of convergence for nonconvex optimization on manifolds,

    N. Boumal, P.-A. Absil, and C. Cartis, “Global rates of convergence for nonconvex optimization on manifolds,”IMA J. Numer . Anal., vol. 39, no. 1, pp. 1–33, 2019

    work page 2019

  44. [44]

    Semidefinite relaxation of quadratic optimization problems,

    Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y . Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,”IEEE Signal Process. Mag., vol. 27, no. 3, pp. 20–34, 2010

    work page 2010

  45. [45]

    Complex-valued matrix differentiation: Techniques and key results,

    A. Hjorungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,”IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2740–2746, 2007

    work page 2007