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January  2020, 16(1): 431-443. doi: 10.3934/jimo.2018161

A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK

* Corresponding author: Stéphane Chrétien

Received  June 2017 Revised  June 2018 Published  November 2018

State estimation in power grids is a crucial step for monitoring and control tasks. It was shown that the state estimation problem can be solved using a convex relaxation based on semi-definite programming. In the present paper, we propose a fast algorithm for solving this relaxation. Our approach uses the Bürer Monteiro factorisation is a special way that solves the problem on the sphere and and estimates the scale in a Gauss-Seidel fashion. Simulations results confirm the promising behavior of the method.

Citation: Stephane Chretien, Paul Clarkson. A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids. Journal of Industrial & Management Optimization, 2020, 16 (1) : 431-443. doi: 10.3934/jimo.2018161
References:
[1]

H. BaiG. LiS. LiQ. LiQ. Jiang and L. Chang, Alternating optimization of sensing matrix and sparsifying dictionary for compressed sensing, IEEE Transactions on Signal Processing, 63 (2015), 1581-1594.  doi: 10.1109/TSP.2015.2399864.  Google Scholar

[2]

R. G. Baraniuk, Compressive sensing [lecture notes], IEEE Signal Processing Magazine, 24 (2007), 118-121.   Google Scholar

[3]

A. Belloni, V. Chernozhukov, L. Wang, et al., Pivotal estimation via square-root lasso in nonparametric regression, The Annals of Statistics, 42 (2014), 757-788. doi: 10.1214/14-AOS1204.  Google Scholar

[4]

S. Bhojanapalli, B. Neyshabur and N. Srebro, Global optimality of local search for low rank matrix recovery, arXiv: 1605.07221, 2016. Google Scholar

[5]

D. Bienstock and G. Munoz, Lp formulations for mixed-integer polynomial optimization problems, arXiv Preprint, 2015. doi: 10.1137/15M1054079.  Google Scholar

[6]

J.-F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Springer Science & Business Media, 2006.  Google Scholar

[7]

N. Boumal, V. Voroninski and A. S. Bandeira, The non-convex burer-monteiro approach works on smooth semidefinite programs, arXiv: 1606.04970, 2016. Google Scholar

[8]

S. Burer and R. D. C. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Mathematical Programming, 95 (2003), 329-357.  doi: 10.1007/s10107-002-0352-8.  Google Scholar

[9]

J.-F. CaiE. J. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.  Google Scholar

[10]

E. J. Candes, The restricted isometry property and its implications for compressed sensing, Comptes Rendus Mathematique, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[11]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5.  Google Scholar

[12]

E. J. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2005), 4203-4215.  doi: 10.1109/TIT.2005.858979.  Google Scholar

[13]

E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2010), 2053-2080.  doi: 10.1109/TIT.2010.2044061.  Google Scholar

[14]

E. J. Candès and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 21-30.   Google Scholar

[15]

S. Chrétien, An alternating l_1 approach to the compressed sensing problem, IEEE Signal Processing Letters, 17 (2010), 181-184.   Google Scholar

[16]

S. Chrétien and S. Darses, Sparse recovery with unknown variance: A lasso-type approach, IEEE Transactions on Information Theory, 60 (2014), 3970-3988.  doi: 10.1109/TIT.2014.2301162.  Google Scholar

[17]

S. Chrétien and T. Wei, Sensing tensors with gaussian filters, IEEE Transactions on Information Theory, 63 (2017), 843-852.  doi: 10.1109/TIT.2016.2633413.  Google Scholar

[18]

M. A. Davenport and J. Romberg, An overview of low-rank matrix recovery from incomplete observations, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 608-622.   Google Scholar

[19]

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, 2012. doi: 10.1515/dmvm-2014-0014.  Google Scholar

[20]

G. Fazelnia, R. Madani and J. Lavaei, Convex relaxation for optimal distributed control problem, in 53rd IEEE Conference on Decision and Control, IEEE, 2014,896-903. doi: 10.1109/TCNS.2014.2309732.  Google Scholar

[21]

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhäuser Basel, 2013. doi: 10.1007/978-0-8176-4948-7.  Google Scholar

[22]

R. Ge, C. Jin and Y. Zheng, No spurious local minima in nonconvex low rank problems: A unified geometric analysis, arXiv: 1704.00708, 2017. Google Scholar

[23]

C. GiraudS. Huet and N. Verzelen, High-dimensional regression with unknown variance, Statistical Science, (2012), 500-518.  doi: 10.1214/12-STS398.  Google Scholar

[24]

R. A. Jabr, Exploiting sparsity in sdp relaxations of the opf problem, IEEE Transactions on Power Systems, 2 (2012), 1138-1139.   Google Scholar

[25]

C. Klauber and H. Zhu, Distribution system state estimation using semidefinite programming, in North American Power Symposium (NAPS), 2015, IEEE, 2015, 1-6. Google Scholar

[26]

O. Klopp and S. Gaiffas, High dimensional matrix estimation with unknown variance of the noise, arXiv: 1112.3055, 2011. Google Scholar

[27]

G. Kutyniok, Theory and applications of compressed sensing, GAMM-Mitteilungen, 36 (2013), 79-101.  doi: 10.1002/gamm.201310005.  Google Scholar

[28]

J. Lavaei and S. H. Low, Zero duality gap in optimal power flow problem, IEEE Transactions on Power Systems, 27 (2012), 92-107.   Google Scholar

[29]

Q. Li and G. Tang, The nonconvex geometry of low-rank matrix optimizations with general objective functions, arXiv: 1611.03060, 2016. Google Scholar

[30]

S. H. Low, Convex relaxation of optimal power flow, part ⅱ: Exactness, arXiv: 1405.0814, 2014. doi: 10.1109/TCNS.2014.2323634.  Google Scholar

[31]

R. Madani, J. Lavaei and R. Baldick, Convexification of power flow equations for power systems in presence of noisy measurements, preprint, 2016. Google Scholar

[32]

D. K. MolzahnJ. T. HolzerB. C. Lesieutre and C. L. DeMarco, Implementation of a large-scale optimal power flow solver based on semidefinite programming, IEEE Transactions on Power Systems, 28 (2013), 3987-3998.   Google Scholar

[33]

J. Nocedal and S. Wright, Numerical Optimization, Springer Science & Business Media, 2006.  Google Scholar

[34]

D. Park, A. Kyrillidis, C. Caramanis and S. Sanghavi, Non-square matrix sensing without spurious local minima via the burer-monteiro approach, arXiv: 1609.03240, 2016. Google Scholar

[35]

J. Salmon, On High Dimensional Regression: Computational and Statistical Perspectives, PhD thesis, HDR, École normale supérieure Paris-Saclay, 2017. Google Scholar

[36]

F. Schweppe, Recursive state estimation: unknown but bounded errors and system inputs, IEEE Transactions on Automatic Control, 13 (1968), 22-28.   Google Scholar

[37]

Q. Song, H. Ge, J. Caverlee and X. Hu, Tensor completion algorithms in big data analytics, arXiv: 1711.10105, 2017. Google Scholar

[38]

A. Virouleau, A. Guilloux, S. Gaïffas and M. Bogdan, High-dimensional robust regression and outliers detection with slope, arXiv: 1712.02640, 2017. Google Scholar

[39]

A. Wang and Z. Jin, Near-optimal noisy low-tubal-rank tensor completion via singular tube thresholding, in Data Mining Workshops (ICDMW), 2017 IEEE International Conference on, IEEE, 2017,553-560. Google Scholar

[40]

F. F. Wu, Power system state estimation: A survey, International Journal of Electrical Power & Energy Systems, 12 (1990), 80-87.   Google Scholar

[41]

Y. Zhang, R. Madani and J. Lavaei, Power system state estimation with line measurements, 2016. Google Scholar

[42]

Z. Zhang and S. Aeron, Exact tensor completion using t-svd, IEEE Transactions on Signal Processing, 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.  Google Scholar

[43]

H. Zhu and G. B. Giannakis, Power system nonlinear state estimation using distributed semidefinite programming, IEEE Journal of Selected Topics in Signal Processing, 8 (2014), 1039-1050.   Google Scholar

[44]

Z. Zhu, Q. Li, G. Tang and M. B. Wakin, The global optimization geometry of nonsymmetric matrix factorization and sensing, arXiv: 1703.01256, 2017. Google Scholar

[45]

R. D. Zimmerman, C. E. Murillo-Sánchez and D. Gan, Matpower, PSERC.[Online]. Software Available at: http://www.pserc.cornell.edu/matpower, 1997. Google Scholar

show all references

References:
[1]

H. BaiG. LiS. LiQ. LiQ. Jiang and L. Chang, Alternating optimization of sensing matrix and sparsifying dictionary for compressed sensing, IEEE Transactions on Signal Processing, 63 (2015), 1581-1594.  doi: 10.1109/TSP.2015.2399864.  Google Scholar

[2]

R. G. Baraniuk, Compressive sensing [lecture notes], IEEE Signal Processing Magazine, 24 (2007), 118-121.   Google Scholar

[3]

A. Belloni, V. Chernozhukov, L. Wang, et al., Pivotal estimation via square-root lasso in nonparametric regression, The Annals of Statistics, 42 (2014), 757-788. doi: 10.1214/14-AOS1204.  Google Scholar

[4]

S. Bhojanapalli, B. Neyshabur and N. Srebro, Global optimality of local search for low rank matrix recovery, arXiv: 1605.07221, 2016. Google Scholar

[5]

D. Bienstock and G. Munoz, Lp formulations for mixed-integer polynomial optimization problems, arXiv Preprint, 2015. doi: 10.1137/15M1054079.  Google Scholar

[6]

J.-F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Springer Science & Business Media, 2006.  Google Scholar

[7]

N. Boumal, V. Voroninski and A. S. Bandeira, The non-convex burer-monteiro approach works on smooth semidefinite programs, arXiv: 1606.04970, 2016. Google Scholar

[8]

S. Burer and R. D. C. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Mathematical Programming, 95 (2003), 329-357.  doi: 10.1007/s10107-002-0352-8.  Google Scholar

[9]

J.-F. CaiE. J. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.  doi: 10.1137/080738970.  Google Scholar

[10]

E. J. Candes, The restricted isometry property and its implications for compressed sensing, Comptes Rendus Mathematique, 346 (2008), 589-592.  doi: 10.1016/j.crma.2008.03.014.  Google Scholar

[11]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5.  Google Scholar

[12]

E. J. Candes and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2005), 4203-4215.  doi: 10.1109/TIT.2005.858979.  Google Scholar

[13]

E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2010), 2053-2080.  doi: 10.1109/TIT.2010.2044061.  Google Scholar

[14]

E. J. Candès and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 21-30.   Google Scholar

[15]

S. Chrétien, An alternating l_1 approach to the compressed sensing problem, IEEE Signal Processing Letters, 17 (2010), 181-184.   Google Scholar

[16]

S. Chrétien and S. Darses, Sparse recovery with unknown variance: A lasso-type approach, IEEE Transactions on Information Theory, 60 (2014), 3970-3988.  doi: 10.1109/TIT.2014.2301162.  Google Scholar

[17]

S. Chrétien and T. Wei, Sensing tensors with gaussian filters, IEEE Transactions on Information Theory, 63 (2017), 843-852.  doi: 10.1109/TIT.2016.2633413.  Google Scholar

[18]

M. A. Davenport and J. Romberg, An overview of low-rank matrix recovery from incomplete observations, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 608-622.   Google Scholar

[19]

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, 2012. doi: 10.1515/dmvm-2014-0014.  Google Scholar

[20]

G. Fazelnia, R. Madani and J. Lavaei, Convex relaxation for optimal distributed control problem, in 53rd IEEE Conference on Decision and Control, IEEE, 2014,896-903. doi: 10.1109/TCNS.2014.2309732.  Google Scholar

[21]

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Birkhäuser Basel, 2013. doi: 10.1007/978-0-8176-4948-7.  Google Scholar

[22]

R. Ge, C. Jin and Y. Zheng, No spurious local minima in nonconvex low rank problems: A unified geometric analysis, arXiv: 1704.00708, 2017. Google Scholar

[23]

C. GiraudS. Huet and N. Verzelen, High-dimensional regression with unknown variance, Statistical Science, (2012), 500-518.  doi: 10.1214/12-STS398.  Google Scholar

[24]

R. A. Jabr, Exploiting sparsity in sdp relaxations of the opf problem, IEEE Transactions on Power Systems, 2 (2012), 1138-1139.   Google Scholar

[25]

C. Klauber and H. Zhu, Distribution system state estimation using semidefinite programming, in North American Power Symposium (NAPS), 2015, IEEE, 2015, 1-6. Google Scholar

[26]

O. Klopp and S. Gaiffas, High dimensional matrix estimation with unknown variance of the noise, arXiv: 1112.3055, 2011. Google Scholar

[27]

G. Kutyniok, Theory and applications of compressed sensing, GAMM-Mitteilungen, 36 (2013), 79-101.  doi: 10.1002/gamm.201310005.  Google Scholar

[28]

J. Lavaei and S. H. Low, Zero duality gap in optimal power flow problem, IEEE Transactions on Power Systems, 27 (2012), 92-107.   Google Scholar

[29]

Q. Li and G. Tang, The nonconvex geometry of low-rank matrix optimizations with general objective functions, arXiv: 1611.03060, 2016. Google Scholar

[30]

S. H. Low, Convex relaxation of optimal power flow, part ⅱ: Exactness, arXiv: 1405.0814, 2014. doi: 10.1109/TCNS.2014.2323634.  Google Scholar

[31]

R. Madani, J. Lavaei and R. Baldick, Convexification of power flow equations for power systems in presence of noisy measurements, preprint, 2016. Google Scholar

[32]

D. K. MolzahnJ. T. HolzerB. C. Lesieutre and C. L. DeMarco, Implementation of a large-scale optimal power flow solver based on semidefinite programming, IEEE Transactions on Power Systems, 28 (2013), 3987-3998.   Google Scholar

[33]

J. Nocedal and S. Wright, Numerical Optimization, Springer Science & Business Media, 2006.  Google Scholar

[34]

D. Park, A. Kyrillidis, C. Caramanis and S. Sanghavi, Non-square matrix sensing without spurious local minima via the burer-monteiro approach, arXiv: 1609.03240, 2016. Google Scholar

[35]

J. Salmon, On High Dimensional Regression: Computational and Statistical Perspectives, PhD thesis, HDR, École normale supérieure Paris-Saclay, 2017. Google Scholar

[36]

F. Schweppe, Recursive state estimation: unknown but bounded errors and system inputs, IEEE Transactions on Automatic Control, 13 (1968), 22-28.   Google Scholar

[37]

Q. Song, H. Ge, J. Caverlee and X. Hu, Tensor completion algorithms in big data analytics, arXiv: 1711.10105, 2017. Google Scholar

[38]

A. Virouleau, A. Guilloux, S. Gaïffas and M. Bogdan, High-dimensional robust regression and outliers detection with slope, arXiv: 1712.02640, 2017. Google Scholar

[39]

A. Wang and Z. Jin, Near-optimal noisy low-tubal-rank tensor completion via singular tube thresholding, in Data Mining Workshops (ICDMW), 2017 IEEE International Conference on, IEEE, 2017,553-560. Google Scholar

[40]

F. F. Wu, Power system state estimation: A survey, International Journal of Electrical Power & Energy Systems, 12 (1990), 80-87.   Google Scholar

[41]

Y. Zhang, R. Madani and J. Lavaei, Power system state estimation with line measurements, 2016. Google Scholar

[42]

Z. Zhang and S. Aeron, Exact tensor completion using t-svd, IEEE Transactions on Signal Processing, 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.  Google Scholar

[43]

H. Zhu and G. B. Giannakis, Power system nonlinear state estimation using distributed semidefinite programming, IEEE Journal of Selected Topics in Signal Processing, 8 (2014), 1039-1050.   Google Scholar

[44]

Z. Zhu, Q. Li, G. Tang and M. B. Wakin, The global optimization geometry of nonsymmetric matrix factorization and sensing, arXiv: 1703.01256, 2017. Google Scholar

[45]

R. D. Zimmerman, C. E. Murillo-Sánchez and D. Gan, Matpower, PSERC.[Online]. Software Available at: http://www.pserc.cornell.edu/matpower, 1997. Google Scholar

Figure 1.  Comparison of Sum of Squared Errors for the IEEE-30 network: New method vs. SDP relaxation (using YALMIP) with noise standard deviation equal to.2 when power is observed at half the number of buses chosen at random.
Figure 2.  Comparison of computation times for the IEEE-30 network: New method vs. SDP relaxation (using YALMPI) with noise standard deviation equal to.2 when power is observed at half the number of buses chosen uniformly at random.
Figure 3.  Example of evolution of the objective function as a function of iteration number for one realisation of a random noise for the IEEE-30 network.
Figure 4.  Example of evolution of the euclidean distance between successive $A$-iterates as a function of iteration number for one realisation of a random noise for the IEEE-30 network.
Figure 5.  Mean Squared Error obtained using the estimator based on the new method with noise standard deviation equal to.2 when power is observed at half the buses. The buses selected for observation were selected uniformly at random.
Figure 6.  Computation times using the new method with noise standard deviation equal to.2 when power is observed at half the buses. The buses selected for observation were selected uniformly at random.
Table1 
Result: $W_{opt}$
Choose $A^{(1,1)} \in \mathbb C^{n\times k}$
                         $\underline {First\;stage} $
$\begin{array}{l} {\bf{while}}\;s \le S - 1\;{\bf{do}}\\ \;\left| \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\nabla g(A) = 2\;\sum\limits_{l = 1}^L \; ( - {z_l}\;\alpha (H_l^* + {H_l})A\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + 2\;{\alpha ^2}\;{\rm{trace}}({H_l}A{A^*})(H_l^* + {H_l})A).\;\;\;\;\;\;\;\;\;\;\left( 8 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\tilde A}^{(t,s + 1)}} = {A^{(t,s)}} - \eta \nabla g({A^{(t,s)}})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 9 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{A^{(t,s + 1)}} = \frac{1}{{\left\| {{{\tilde A}^{(t,s + 1)}}} \right\|}}\;{{\tilde A}^{(t,s + 1)}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {10} \right) \end{array} \right.\\ {\bf{end}} \end{array}$
Set $A^{(t+1,1)}=A^{(t,S)}$.
                         $\underline {Second\;stage}$
Set
        $\begin{align} W_{opt}&= \alpha^{(t+1)} \ A^{(t+1,1)}A^{(t+1,1)^*} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {11} \right) \end{align}$
with
\begin{align} \alpha^{(t+1)} & = \frac{\sum\nolimits_{l=1}^L z_l\ \textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})}{\sum\nolimits_{l=1}^L \left(\textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})\right)^2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \left({12} \right) \end{align}
Algorithm 1: The two stage optimisation procedure
Result: $W_{opt}$
Choose $A^{(1,1)} \in \mathbb C^{n\times k}$
                         $\underline {First\;stage} $
$\begin{array}{l} {\bf{while}}\;s \le S - 1\;{\bf{do}}\\ \;\left| \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\nabla g(A) = 2\;\sum\limits_{l = 1}^L \; ( - {z_l}\;\alpha (H_l^* + {H_l})A\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + 2\;{\alpha ^2}\;{\rm{trace}}({H_l}A{A^*})(H_l^* + {H_l})A).\;\;\;\;\;\;\;\;\;\;\left( 8 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\tilde A}^{(t,s + 1)}} = {A^{(t,s)}} - \eta \nabla g({A^{(t,s)}})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 9 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{A^{(t,s + 1)}} = \frac{1}{{\left\| {{{\tilde A}^{(t,s + 1)}}} \right\|}}\;{{\tilde A}^{(t,s + 1)}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {10} \right) \end{array} \right.\\ {\bf{end}} \end{array}$
Set $A^{(t+1,1)}=A^{(t,S)}$.
                         $\underline {Second\;stage}$
Set
        $\begin{align} W_{opt}&= \alpha^{(t+1)} \ A^{(t+1,1)}A^{(t+1,1)^*} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {11} \right) \end{align}$
with
\begin{align} \alpha^{(t+1)} & = \frac{\sum\nolimits_{l=1}^L z_l\ \textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})}{\sum\nolimits_{l=1}^L \left(\textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})\right)^2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \left({12} \right) \end{align}
Algorithm 1: The two stage optimisation procedure
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