Abstract

The determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind [ S. Twomey, J. Comput. Phys. 18, 188 (1975); Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007) ], since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature of the problem, regularization techniques were developed. However, most of the literature focuses on the application of Phillips–Twomey regularization and its variants, which are unstable in several cases. As is known, the particle size distribution is always nonnegative, and we are often faced with incomplete data. Therefore, we study the active set method and propose a regularizing active set algorithm for ill-posed particle size distribution function retrieval and for enforcing nonnegativity in computation. Our numerical tests are based on synthetic data for theoretical simulations and the field data obtained with a CE 318 Sun photometer for the Po Yang lake region of Jiang Xi Province, China, and are performed to show the efficiency and feasibility of the proposed algorithms.

© 2008 Optical Society of America

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References

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  1. S. Twomey, Atmospheric Aerosols (Elsevier, 1977).
  2. C. E. Junge, Air Chemistry and Radioactivity (Academic, 1963).
  3. G. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  4. C. N. Davies, "Size distribution of atmospheric aerosol," J. Aerosol Sci. 5, 293-300 (1974).
    [CrossRef]
  5. G. J. Mccartney, Optics of Atmosphere (Wiley, 1976).
  6. A. Ångström, "On the atmospheric transmission of sun radiation and on dust in the air," Geogr. Ann. 11, 156-166 (1929).
    [CrossRef]
  7. G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64, 4-10 (1983).
    [CrossRef]
  8. J. A. Curcio, "Evaluation of atmospheric aerosol particle size distribution from scattering measurements in the visible and infrared," J. Opt. Soc. Am. 51, 548-551 (1961).
    [CrossRef]
  9. D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach. 9, 84-97 (1962).
    [CrossRef]
  10. S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).
    [CrossRef]
  11. G. E. Shaw, "Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra," Appl. Opt. 18, 988-993 (1979).
    [CrossRef] [PubMed]
  12. Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, "Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space," Appl. Opt. 28, 7456-7467 (2006).
    [CrossRef]
  13. K. S. Shifrin and L. G. Zolotov, "Spectral attenuation and aerosol particle size distribution," Appl. Opt. 35, 2114-2124 (1996).
    [CrossRef] [PubMed]
  14. C. Böckmann, "Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions," Appl. Opt. 40, 1329-1342 (2001).
    [CrossRef]
  15. S. Twomey, "Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions," J. Comput. Phys. 18, 188-200 (1975).
    [CrossRef]
  16. Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007).
  17. M. T. Chahine, "Inverse problems in radiation transfer: determination of atmospheric parameters," J. Aerosol Sci. 27, 960-967 (1970).
  18. F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995).
    [CrossRef] [PubMed]
  19. G. Yamamoto and M. Tanaka, "Determination of aerosol size distribution function from spectral attenuation measurements," Appl. Opt. 8, 447-453 (1969).
    [CrossRef] [PubMed]
  20. H. Grassl, "Determination of aerosol size distributions from spectral attenuation measurements," Appl. Opt. 10, 2534-2538 (1971).
    [CrossRef] [PubMed]
  21. K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
    [CrossRef]
  22. C. Böckmann and A. Kirsche, "Iterative regularization method for lidar remote sensing," Comput. Phys. Commun. 174, 607-615 (2006).
    [CrossRef]
  23. A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31, 767-768 (2000).
    [CrossRef]
  24. Y. F. Wang, S. F. Fan, and X. Feng, "Retrieval of the aerosol particle size distribution function by incorporating a priori information," J. Aerosol Sci. 38, 885-901 (2007).
    [CrossRef]
  25. J. Heintzenberg, "Properties of log-normal particle size distributions," Aerosol Sci. Technol. 21, 46-48 (1994).
    [CrossRef]
  26. D. L. Wright, "Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution," J. Aerosol Sci. 31, 1-18 (2000).
    [CrossRef]
  27. M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, and Y. Xu, "Inversion of particle-size distribution from angular light-scattering data with genetic algorithms," Appl. Opt. 38, 2677-2685 (1999).
    [CrossRef]
  28. D. A. Ligon, J. B. Gillespie, and P. Pellegrino, "Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method," Appl. Opt. 39, 4402-4410 (2000).
    [CrossRef]
  29. G. Ramachardran, D. Lieth, and L. Todd, "Extraction of spatial aerosol distribution from multispectral light extinction measurements with computed tomography," J. Opt. Soc. Am. A 11, 144-154 (1994).
    [CrossRef]
  30. T. Nguyen and K. Cox, "A method for the determination of aerosol particle distributions from light extinction data," in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, 1989), pp. 330.
  31. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  32. T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Science, 2003).
  33. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
    [CrossRef]
  34. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, 1994).
    [CrossRef]
  35. Y. X. Yuan, Numerical Methods for Nonlinear Programming (Shanghai Science and Technology, 1993).
  36. J. Nocedal and S. J. Wright, Numerical optimization (Springer, 1999).
    [CrossRef]
  37. R. Fletcher, Pratical Methods of Optimization (Wiley, 1987).
  38. M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
    [CrossRef]
  39. C. E. Junge, "The size distribution and aging of natural aerosols as determined from electrical and optical data on the atmosphere," J. Meteorol. 12, 13-25 (1955).
    [CrossRef]
  40. W. E. Clark and K. T. Whitby, "Concentration and size distribution measurements of atmospheric aerosols and a test of the theory of self-preserving size distributions," J. Atmos. Sci. 24, 677-687 (1967).
    [CrossRef]
  41. A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
    [CrossRef]
  42. M. Weindisch and W. von Hoyningen-Huen, "Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements," Atmos. Environ. 28, 785-792 (1994).
    [CrossRef]
  43. F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).

2007 (1)

Y. F. Wang, S. F. Fan, and X. Feng, "Retrieval of the aerosol particle size distribution function by incorporating a priori information," J. Aerosol Sci. 38, 885-901 (2007).
[CrossRef]

2006 (2)

Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, "Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space," Appl. Opt. 28, 7456-7467 (2006).
[CrossRef]

C. Böckmann and A. Kirsche, "Iterative regularization method for lidar remote sensing," Comput. Phys. Commun. 174, 607-615 (2006).
[CrossRef]

2003 (1)

F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).

2001 (1)

2000 (3)

A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31, 767-768 (2000).
[CrossRef]

D. L. Wright, "Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution," J. Aerosol Sci. 31, 1-18 (2000).
[CrossRef]

D. A. Ligon, J. B. Gillespie, and P. Pellegrino, "Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method," Appl. Opt. 39, 4402-4410 (2000).
[CrossRef]

1999 (1)

1996 (1)

1995 (1)

1994 (4)

J. Heintzenberg, "Properties of log-normal particle size distributions," Aerosol Sci. Technol. 21, 46-48 (1994).
[CrossRef]

G. Ramachardran, D. Lieth, and L. Todd, "Extraction of spatial aerosol distribution from multispectral light extinction measurements with computed tomography," J. Opt. Soc. Am. A 11, 144-154 (1994).
[CrossRef]

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

M. Weindisch and W. von Hoyningen-Huen, "Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements," Atmos. Environ. 28, 785-792 (1994).
[CrossRef]

1983 (1)

G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64, 4-10 (1983).
[CrossRef]

1980 (1)

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

1979 (1)

1978 (1)

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

1975 (1)

S. Twomey, "Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions," J. Comput. Phys. 18, 188-200 (1975).
[CrossRef]

1974 (1)

C. N. Davies, "Size distribution of atmospheric aerosol," J. Aerosol Sci. 5, 293-300 (1974).
[CrossRef]

1971 (1)

1970 (1)

M. T. Chahine, "Inverse problems in radiation transfer: determination of atmospheric parameters," J. Aerosol Sci. 27, 960-967 (1970).

1969 (1)

1967 (1)

W. E. Clark and K. T. Whitby, "Concentration and size distribution measurements of atmospheric aerosols and a test of the theory of self-preserving size distributions," J. Atmos. Sci. 24, 677-687 (1967).
[CrossRef]

1963 (1)

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach. 9, 84-97 (1962).
[CrossRef]

1961 (1)

1955 (1)

C. E. Junge, "The size distribution and aging of natural aerosols as determined from electrical and optical data on the atmosphere," J. Meteorol. 12, 13-25 (1955).
[CrossRef]

1929 (1)

A. Ångström, "On the atmospheric transmission of sun radiation and on dust in the air," Geogr. Ann. 11, 156-166 (1929).
[CrossRef]

Acta Opt. Sin. (1)

F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).

Aerosol Sci. Technol. (1)

J. Heintzenberg, "Properties of log-normal particle size distributions," Aerosol Sci. Technol. 21, 46-48 (1994).
[CrossRef]

Appl. Opt. (9)

M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, and Y. Xu, "Inversion of particle-size distribution from angular light-scattering data with genetic algorithms," Appl. Opt. 38, 2677-2685 (1999).
[CrossRef]

D. A. Ligon, J. B. Gillespie, and P. Pellegrino, "Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method," Appl. Opt. 39, 4402-4410 (2000).
[CrossRef]

G. E. Shaw, "Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra," Appl. Opt. 18, 988-993 (1979).
[CrossRef] [PubMed]

Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, "Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space," Appl. Opt. 28, 7456-7467 (2006).
[CrossRef]

K. S. Shifrin and L. G. Zolotov, "Spectral attenuation and aerosol particle size distribution," Appl. Opt. 35, 2114-2124 (1996).
[CrossRef] [PubMed]

C. Böckmann, "Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions," Appl. Opt. 40, 1329-1342 (2001).
[CrossRef]

F. Ferri, A. Bassini, and E. Paganini, "Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing," Appl. Opt. 34, 5829-5839 (1995).
[CrossRef] [PubMed]

G. Yamamoto and M. Tanaka, "Determination of aerosol size distribution function from spectral attenuation measurements," Appl. Opt. 8, 447-453 (1969).
[CrossRef] [PubMed]

H. Grassl, "Determination of aerosol size distributions from spectral attenuation measurements," Appl. Opt. 10, 2534-2538 (1971).
[CrossRef] [PubMed]

Astrophys. J. (1)

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Atmos. Environ. (1)

M. Weindisch and W. von Hoyningen-Huen, "Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements," Atmos. Environ. 28, 785-792 (1994).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64, 4-10 (1983).
[CrossRef]

Comput. Phys. Commun. (1)

C. Böckmann and A. Kirsche, "Iterative regularization method for lidar remote sensing," Comput. Phys. Commun. 174, 607-615 (2006).
[CrossRef]

Geogr. Ann. (1)

A. Ångström, "On the atmospheric transmission of sun radiation and on dust in the air," Geogr. Ann. 11, 156-166 (1929).
[CrossRef]

J. Aerosol Sci. (5)

M. T. Chahine, "Inverse problems in radiation transfer: determination of atmospheric parameters," J. Aerosol Sci. 27, 960-967 (1970).

D. L. Wright, "Retrieval of optical properties of atmospheric aerosols from moments of the particle size distribution," J. Aerosol Sci. 31, 1-18 (2000).
[CrossRef]

A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31, 767-768 (2000).
[CrossRef]

Y. F. Wang, S. F. Fan, and X. Feng, "Retrieval of the aerosol particle size distribution function by incorporating a priori information," J. Aerosol Sci. 38, 885-901 (2007).
[CrossRef]

C. N. Davies, "Size distribution of atmospheric aerosol," J. Aerosol Sci. 5, 293-300 (1974).
[CrossRef]

J. Assoc. Comput. Mach. (2)

D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach. 9, 84-97 (1962).
[CrossRef]

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).
[CrossRef]

J. Atmos. Sci. (2)

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

W. E. Clark and K. T. Whitby, "Concentration and size distribution measurements of atmospheric aerosols and a test of the theory of self-preserving size distributions," J. Atmos. Sci. 24, 677-687 (1967).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, "Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions," J. Comput. Phys. 18, 188-200 (1975).
[CrossRef]

J. Geophys. Res. (1)

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

J. Meteorol. (1)

C. E. Junge, "The size distribution and aging of natural aerosols as determined from electrical and optical data on the atmosphere," J. Meteorol. 12, 13-25 (1955).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (13)

T. Nguyen and K. Cox, "A method for the determination of aerosol particle distributions from light extinction data," in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, 1989), pp. 330.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Science, 2003).

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Kluwer Academic, 1994).
[CrossRef]

Y. X. Yuan, Numerical Methods for Nonlinear Programming (Shanghai Science and Technology, 1993).

J. Nocedal and S. J. Wright, Numerical optimization (Springer, 1999).
[CrossRef]

R. Fletcher, Pratical Methods of Optimization (Wiley, 1987).

G. J. Mccartney, Optics of Atmosphere (Wiley, 1976).

S. Twomey, Atmospheric Aerosols (Elsevier, 1977).

C. E. Junge, Air Chemistry and Radioactivity (Academic, 1963).

G. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007).

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Figures (4)

Fig. 1
Fig. 1

Input and results retrieved with our inversion method for error level δ = 0.005 and different complex refractive indices.

Fig. 2
Fig. 2

Input and results retrieved with our inversion method for error level δ = 0.01 and different complex refractive indices.

Fig. 3
Fig. 3

Input and results retrieved with our inversion method for error level δ = 0.05 and different complex refractive indices.

Fig. 4
Fig. 4

Particle size distribution in October 2005 (mornings).

Tables (1)

Tables Icon

Table 1 Root Mean-Square Errors for Various Noise Levels δ

Equations (52)

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τ aero ( λ ) = 0 π r 2 Q ex ( r , λ , η ) n ( r ) d r + ϱ ( λ ) ,
K : F O ,
( K n ) ( λ ) + ϱ ( λ ) = 0 k ( r , λ , η ) n ( r ) d r + ϱ ( λ ) = o ( λ ) + ϱ ( λ ) = d ( λ ) ,
K n + ϱ = o + ϱ = d .
min 1 2 K n d 2 + ν Ω [ n ] ,
min J 0 [ n ] 1 2 K n d 2 + ν 2 ( D n , n )
J 0 [ n ] 1 2 [ ( K * K + ν D ) n , n ] ( d , K n ) + 1 2 d 2
min J [ n ] 1 2 [ ( K * K + ν D ) n , n ] ( d , K n )
min J s k [ s k + n k ] = 1 2 ( G s k , s k ) + ( g k , s k ) + c ,
s.t. s k j = 0 , j W k
min Q [ s k ] = 1 2 ( G s k , s k ) + ( g k , s k ) ,
s.t. s k j = 0 , j W k .
α k min { 1 , min j W k , s k j < 0 n k j s k j } .
j W k * λ j * = G n k * K * d ,
G s k * + g k j W k * λ j * = 0 ,
s k * j = 0 , j W k * ,
λ j * 0 , j W k * .
G n k * K * d j W k λ j * = 0 ,
n k * j = 0 , j W k ,
n k * j 0 , j S , j W k ,
D = [ 1 + 1 h r 2 1 h r 2 0 0 1 h r 2 1 + 2 h r 2 1 h r 2 0 0 1 h r 2 1 + 2 h r 2 1 h r 2 0 0 1 h r 2 1 + 1 h r 2 ]
ν k = ν 0 ξ k 1 ,
τ aero ( λ ) = a b [ k ( r , λ , η ) h ( r ) ] f ( r ) d r ,
( Ξ f ) ( λ ) = τ aero ( λ ) .
K f = τ aero .
min 1 2 K f τ aero 2 + ν D 1 2 f 2
d = o + δ × rand ( size ( o ) ) ,
rmse = 1 m i = 1 m [ τ comp ( λ i ) τ meas ( λ i ) ] 2 [ τ comp ( λ i ) ] 2 ,
n true ( r ) = 10.5 r 3.5 exp ( 10 12 r 2 ) .
h ( r ) = C r ( ν * + 1 ) ,
min Q ( x ) = 1 2 x T G x p T x ,
s.t. a i T x = b i , i E ,
a i T x b i , i I ,
L ( x , λ ) = Q ( x ) i E cup I λ i ( a i T x b i ) .
S ( x * ) = { i E I : a i T x * = b i } ,
G x * p i S ( x * ) λ i * a i = 0 ,
a i T x * = b i , for all i S ( x * ) ,
a i T x * b i , for all i I \ S ( x * ) ,
λ i * 0 , for all i I S ( x * ) .
min Q ( x ) = 1 2 x T G x p T x ,
s.t. A x = b ,
L ( x , λ ) = Q ( x ) λ T ( A x b ) .
[ G A T A 0 ] [ x * λ * ] = [ p b ] .
min J [ n ] , s.t. n S .
min Q [ s k ] , s.t. s k j = 0 , j W k .
ρ 0 z 0 T z 0 and set k 1 .
α k ρ k 1 [ s k 1 T ( G s k 1 ) ] ,
s k s k 1 + α k z k 1 ,
grad k [ Q ] grad k 1 [ Q ] + α k G s k ( such that grad k [ Q ] is a feasible direction ) ,
ρ k grad k [ Q ] T grad k [ Q ] ,
β k ρ k ρ k 1 ,
z k grad k 1 [ Q ] + β k z k 1 .

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