References & Further Reading
References
- Q. Wu and R. Zhang, Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming, 2019
The first RIS paper to use SDR and element-wise BCD. Algorithm 3 and Proposition 2 from this paper are Section 6.4.
- Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, Semidefinite relaxation of quadratic optimization problems, 2010
The tutorial reference for SDR. Everything in Section 6.2 follows the structure laid out here.
- S. Zhang and Y. Huang, Complex quadratic optimization and semidefinite programming, 2006
Derivation of the $(\pi/4)$ approximation ratio for complex-valued SDR. Theorem 6.6 rests on this paper.
- A. M.-C. So, J. Zhang, and Y. Ye, On approximating complex quadratic optimization problems via semidefinite programming relaxations, 2007
Sharper approximation-ratio analysis for complex QCQP. Develops the randomization procedure used in Algorithm 6.1.
- X. Yu, D. Xu, Y. Sun, D. W. K. Ng, and R. Schober, Robust and Secure Wireless Communications via Intelligent Reflecting Surfaces, 2020
Introduces manifold optimization for RIS. Section III and Algorithm 1 are directly reproduced as Section 6.3.
- P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008
The standard reference for Riemannian optimization. Chapter 4 covers tangent spaces and gradients; Chapters 5–7 develop algorithms. Companion website has MATLAB code.
- N. Boumal, An Introduction to Optimization on Smooth Manifolds, Cambridge University Press, 2023
Modern successor to Absil et al. 2008; more accessible treatment. Companion to Manopt toolbox.
- D. P. Bertsekas, Nonlinear Programming, Athena Scientific, 2nd ed., 1999
Block coordinate descent convergence theory. Section 2.7 is referenced throughout this chapter.
- C. Pan et al., Multicell MIMO Communications Relying on Intelligent Reflecting Surfaces, 2020
Multi-user formulation with SDR. The rank-$r$ multi-user objective (Eq. 16) is from this paper.
- M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, 2014. [Link]
The CVX toolbox used by nearly every SDR-based RIS paper. Excellent for prototyping but slow at large $N$.
- M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, 1995
The originating paper of the SDR + randomization paradigm for MAX-CUT. Set the template for all subsequent SDR approximation work.
- G. Caire and I. Atzeni, Fast Beam Optimization for Array-Fed RIS via Riemannian BCD, 2023
CommIT contribution. Fast manifold-BCD hybrid for array-fed RIS — key algorithmic enabler for real-time optimization at $N \geq 2048$.
Further Reading
Resources for deeper study of the algorithmic landscape.
Riemannian optimization in MATLAB / Python
N. Boumal et al., 'Manopt: a Matlab toolbox for optimization on manifolds,' JMLR, 2014
The Manopt toolbox is the production-quality implementation of manifold methods. Its documentation is the best hands-on introduction.
Complex-variable SDP with applications
S. Boyd and L. Vandenberghe, 'Semidefinite Programming,' SIAM Review, 1996
Classical survey of SDP, with applications including quadratic optimization. Important for understanding why SDR works where it does.
Low-rank SDP: fast algorithms
S. Burer and R. D. C. Monteiro, 'A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization,' Math. Programming, 2003
Burer-Monteiro factorization $\mathbf{X} = \mathbf{Y}\mathbf{Y}^H$ bypasses the interior-point bottleneck for low-rank solutions. Relevant for large-$N$ SDR.
Manifold optimization for wireless
Alexandropoulos, Islam, and Xenakis (2023), 'Reconfigurable Intelligent Surfaces: Present Status and Future Perspectives,' IEEE Wireless Commun.
Recent survey with emphasis on manifold methods and their role in real-time RIS optimization.
Element-wise updates with discrete phases
Wu and Zhang (2020), 'Discrete phase shift' paper (Ch. 2 reference)
Extends element-wise BCD to the discrete-phase case (Chapter 8 of this book). Shows that the same framework handles both continuous and discrete constraints.