Abstract

Five search algorithms from the literature of black-box optimization were implemented and applied to optical design problems. These algorithms are the Particle Swarm Optimization, Gravity Search Algorithm, Cuckoo Search, Covariance Matrix Adaptation Evolution Strategy and Nelder&Mead Simplex search. The performance of these search algorithms’ implementations was assessed using the BBOB2009 (Black Box Optimization Benchmark) benchmark suite. These algorithms were compared in the context of two optical case studies, one with conventional rotationally symmetric optics and one with freeform optics. A comparison was performed against a commercial optical design software. Finally we provided a simple restart scheme applicable in the workflow of an optical designer. To the best of our knowledge, this is the first in-depth quantitative comparison of optimization algorithms for optical design.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. Glatzel and R. Wilson, “Adaptive automatic correction in optical design,” Appl. Opt. 7(2), 265–276 (1968).
    [Crossref] [PubMed]
  2. M. J. Kidger, “Use of the Levenberg-Marquardt (damped least-squares) optimization method in lens design,” Opt. Eng. 32, 1731–1740 (1993).
    [Crossref]
  3. D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
    [Crossref]
  4. K. Höschel and L. Vasudevan, “Genetic algorithms for lens design: a review,” J. Opt. 48(1), 134–144 (2019).
    [Crossref]
  5. S. Banerjee and L. Hazra, “Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets,” Appl. Opt. 40(34), 6265–6273 (2001).
    [Crossref] [PubMed]
  6. L. Hazra and B. Saswatee, “Genetic algorithm in the structural design of Cooke triplet lenses,” Proc. SPIE 3737, 172–180 (1999).
    [Crossref]
  7. X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
    [Crossref]
  8. J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).
  9. Z. Shen, J. Yu, Z. Song, L. Chen, Q. Yuan, Z. Gao, S. Pei, B. Liu, and J. Ye, “Customized design and efficient fabrication of two freeform aluminum mirrors by single point diamond turning technique,” Appl. Opt. 58(9), 2269–2276 (2019).
    [Crossref] [PubMed]
  10. Q. Meng, H. Wang, W. Liang, Z. Yan, and B. Wang, “Design of off-axis three-mirror systems with ultrawide field of view based on an expansion process of surface freeform and field of view,” Appl. Opt. 58(3), 609–615 (2019).
    [Crossref] [PubMed]
  11. Z. Li, X. Liu, F. Fang, X. Zhang, Z. Zeng, L. Zhu, and N. Yan, “Integrated manufacture of a freeform off-axis multi-reflective imaging system without optical alignment,” Opt. Express 26(6), 7625–7637 (2018).
    [Crossref] [PubMed]
  12. J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
    [Crossref]
  13. A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt. 51(15), 3054–3058 (2012).
    [Crossref] [PubMed]
  14. A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).
  15. A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).
  16. M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
  17. C. Menke, “Application of particle swarm optimization to the automatic design of optical systems,” Proc. SPIE 10690, 106901A (2018).
    [Crossref]
  18. D. Vasiljevic, Classical and Evolutionary Algorithms in the Optimization of Optical Systems (Springer Science & Business Media, 2012).
  19. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
    [Crossref] [PubMed]
  20. J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
    [Crossref]
  21. T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
    [Crossref] [PubMed]
  22. L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
    [Crossref]
  23. R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science (1995), pp. 39–43.
    [Crossref]
  24. C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
    [Crossref]
  25. T. Zeugmann, P., Poupart and J. Kennedy, “Particle swarm optimization,” in Encyclopedia of Machine Learning, C. Sammut and G. I. Webb, ed. (Springer Science & Business Media, 2011).
  26. E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
    [Crossref]
  27. S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).
  28. X.-S. Yang and D. Suash, “Engineering optimisation by cuckoo search,” Math. Model. Num. Opt. 1(4), 330–343 (2010).
    [Crossref]
  29. A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
    [Crossref]
  30. X.-S. Yang and D. Suash, “Cuckoo search: recent advances and applications,” Neural Comput. Appl. 24, 169–174 (2014).
    [Crossref]
  31. G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
    [Crossref] [PubMed]
  32. N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
    [Crossref] [PubMed]
  33. “The CMA evolution strategy: A tutorial,” 2016, https://arxiv.org/pdf/1604.00772.pdf .
  34. N. Hansen, “Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed,” Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers (2009), pp. 2389–2396.
    [Crossref]
  35. A. Auger and H. Nikolaus, “A restart CMA evolution strategy with increasing population size,” 2005 IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 1769–1776.
    [Crossref]
  36. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
    [Crossref]
  37. A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).
  38. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd edition: The Art of Scientific Computing (Cambridge University, 2007).
  39. N. Hansen, F. Steffen, R. Raymond, and A. Auger, “Real-parameter black-box optimization benchmarking 2010: Experimental setup,” Research report RR-7215, INRIA, 2010.
  40. COCO, (COmparing Continuous Optimisers) homepage, (INRIA, 2019), http://coco.gforge.inria.fr/ .
  41. N. Hansen, F. Steffen, R. Raymond, and A. Auger, “Real-parameter black-box optimization benchmarking 2009: Noiseless functions definitions,” Research report RR-6829, INRIA, 2009.
  42. N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
    [Crossref]
  43. D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1, 67–82 (1997).
    [Crossref]
  44. A. Hagg, A. Asteroth, and T. Bäck, “Prototype discovery using quality-diversity,” International Conference on Parallel Problem Solving from Nature (2018), pp. 500–511.
    [Crossref]
  45. Steel Bank Common Lisp (SBCL) homepage, (2019), http://www.sbcl.org/ .

2019 (3)

2018 (6)

Z. Li, X. Liu, F. Fang, X. Zhang, Z. Zeng, L. Zhu, and N. Yan, “Integrated manufacture of a freeform off-axis multi-reflective imaging system without optical alignment,” Opt. Express 26(6), 7625–7637 (2018).
[Crossref] [PubMed]

C. Menke, “Application of particle swarm optimization to the automatic design of optical systems,” Proc. SPIE 10690, 106901A (2018).
[Crossref]

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
[Crossref]

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).

2017 (2)

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
[Crossref] [PubMed]

2016 (3)

L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
[Crossref]

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

2015 (2)

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

2014 (2)

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

X.-S. Yang and D. Suash, “Cuckoo search: recent advances and applications,” Neural Comput. Appl. 24, 169–174 (2014).
[Crossref]

2013 (1)

A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
[Crossref]

2012 (1)

2010 (1)

X.-S. Yang and D. Suash, “Engineering optimisation by cuckoo search,” Math. Model. Num. Opt. 1(4), 330–343 (2010).
[Crossref]

2009 (1)

E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
[Crossref]

2006 (1)

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

2004 (1)

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

2001 (2)

1999 (1)

L. Hazra and B. Saswatee, “Genetic algorithm in the structural design of Cooke triplet lenses,” Proc. SPIE 3737, 172–180 (1999).
[Crossref]

1997 (1)

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1, 67–82 (1997).
[Crossref]

1994 (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
[Crossref]

1993 (1)

M. J. Kidger, “Use of the Levenberg-Marquardt (damped least-squares) optimization method in lens design,” Opt. Eng. 32, 1731–1740 (1993).
[Crossref]

1968 (1)

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Alavi, A. H.

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
[Crossref]

Alexandre, K.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Antonios, T.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Asteroth, A.

A. Hagg, A. Asteroth, and T. Bäck, “Prototype discovery using quality-diversity,” International Conference on Parallel Problem Solving from Nature (2018), pp. 500–511.
[Crossref]

Auger, A.

A. Auger and H. Nikolaus, “A restart CMA evolution strategy with increasing population size,” 2005 IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 1769–1776.
[Crossref]

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

Bäck, T.

A. Hagg, A. Asteroth, and T. Bäck, “Prototype discovery using quality-diversity,” International Conference on Parallel Problem Solving from Nature (2018), pp. 500–511.
[Crossref]

Banerjee, S.

Bauer, A.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

Bharathwaj, N.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Broemel, A.

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Celal, Y.

S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).

Chen, L.

Cheng, X.

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

Davis, G. E.

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

Dejan, A. G.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Eberhart, R.

R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science (1995), pp. 39–43.
[Crossref]

Fang, F.

Finck, S.

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

Frédéric, G.

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

Fuerschbach, K.

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

Gandomi, A. H.

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
[Crossref]

Gao, Z.

Glatzel, E.

Gross, H.

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Guofan, J.

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Hagg, A.

A. Hagg, A. Asteroth, and T. Bäck, “Prototype discovery using quality-diversity,” International Conference on Parallel Problem Solving from Nature (2018), pp. 500–511.
[Crossref]

Hansen, N.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
[Crossref] [PubMed]

N. Hansen, “Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed,” Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers (2009), pp. 2389–2396.
[Crossref]

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

Hasan, T.

S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).

Hazra, L.

Hongjie, F.

L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
[Crossref]

Höschel, K.

K. Höschel and L. Vasudevan, “Genetic algorithms for lens design: a review,” J. Opt. 48(1), 134–144 (2019).
[Crossref]

Hossein, N.-P.

E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
[Crossref]

Howard, J. M.

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
[Crossref]

Huang, W.

L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
[Crossref]

Hyo-Sang, S.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Jayao, L.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Jin, G. F.

T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
[Crossref] [PubMed]

Juan Carlos, M.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Jun, L. I.

L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
[Crossref]

Kennedy, J.

R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science (1995), pp. 39–43.
[Crossref]

Kidger, M. J.

M. J. Kidger, “Use of the Levenberg-Marquardt (damped least-squares) optimization method in lens design,” Opt. Eng. 32, 1731–1740 (1993).
[Crossref]

Leboucher, C.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Li, Z.

Liang, W.

Lippmann, U.

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Liu, B.

Liu, C.

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

Liu, X.

Luersen, A.

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

Ma, C.

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Macready, W. G.

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1, 67–82 (1997).
[Crossref]

Marco, L. R.

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

Masaki, I.

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

Mathias, F.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Meng, Q.

Menke, C.

C. Menke, “Application of particle swarm optimization to the automatic design of optical systems,” Proc. SPIE 10690, 106901A (2018).
[Crossref]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Nikolaus, H.

A. Auger and H. Nikolaus, “A restart CMA evolution strategy with increasing population size,” 2005 IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 1769–1776.
[Crossref]

Nikolic, M. I.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Ochse, D.

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Oleszko, M.

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Ostermeier, A.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
[Crossref] [PubMed]

Özyön, S.

S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).

Pablo, B.

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Papa, J. C.

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
[Crossref]

Patrick, S.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Pei, S.

Pošík, P.

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

Qun, H.

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

Rachid, C.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Rashedi, E.

E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
[Crossref]

Rodolphe,

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

Rolland, J. P.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
[Crossref]

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

Ros, R.

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

Saeid, S.

E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
[Crossref]

Saswatee, B.

L. Hazra and B. Saswatee, “Genetic algorithm in the structural design of Cooke triplet lenses,” Proc. SPIE 3737, 172–180 (1999).
[Crossref]

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

Shafer, D.

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
[Crossref]

Shen, Z.

Song, Z.

Stéphane, L. M.

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Suash, D.

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

X.-S. Yang and D. Suash, “Cuckoo search: recent advances and applications,” Neural Comput. Appl. 24, 169–174 (2014).
[Crossref]

X.-S. Yang and D. Suash, “Engineering optimisation by cuckoo search,” Math. Model. Num. Opt. 1(4), 330–343 (2010).
[Crossref]

Thompson, K. P.

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

Vasudevan, L.

K. Höschel and L. Vasudevan, “Genetic algorithms for lens design: a review,” J. Opt. 48(1), 134–144 (2019).
[Crossref]

Wang, B.

Wang, G.-G.

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

Wang, H.

Wei, H.

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Wilson, R.

Wolpert, D. H.

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1, 67–82 (1997).
[Crossref]

Xiaodong, Z.

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Yabe, A.

Yan, N.

Yan, X.-S.

A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
[Crossref]

Yan, Z.

Yang, T.

T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
[Crossref] [PubMed]

Yang, X.-S.

X.-S. Yang and D. Suash, “Cuckoo search: recent advances and applications,” Neural Comput. Appl. 24, 169–174 (2014).
[Crossref]

X.-S. Yang and D. Suash, “Engineering optimisation by cuckoo search,” Math. Model. Num. Opt. 1(4), 330–343 (2010).
[Crossref]

Ye, J.

Yongtian, W.

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

Yu, J.

Yuan, Q.

Zeng, Z.

Zhang, X.

Zhang, Y.

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

Zhaojun, Z.

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

Zhong, Y.

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

Zhu, J.

T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
[Crossref] [PubMed]

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Zhu, L.

Adv. Opt. Technol. (1)

A. Broemel, C. Liu, Y. Zhong, Y. Zhang, and H. Gross, “Freeform surface descriptions. Part II: application benchmark,” Adv. Opt. Technol. 6(15), 337–347 (2017).

Appl. Opt. (5)

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Comput. Phys. (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
[Crossref]

Eng. Comput. (1)

A. H. Gandomi, X.-S. Yan, and A. H. Alavi, “Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems,” Eng. Comput. 29, 17–35 (2013).
[Crossref]

Evol. Comput. (1)

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
[Crossref] [PubMed]

IEEE Trans. Evol. Comput. (1)

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1, 67–82 (1997).
[Crossref]

IEEE Trans. Games (1)

C. Leboucher, S. Hyo-Sang, C. Rachid, L. M. Stéphane, S. Patrick, F. Mathias, T. Antonios, and K. Alexandre, “An Enhanced Particle Swarm Optimization Method Integrated With Evolutionary Game Theory,” IEEE Trans. Games 10, 221–230 (2018).
[Crossref]

Inf. Sci. (1)

E. Rashedi, N.-P. Hossein, and S. Saeid, “GSA: a gravitational search algorithm,” Inf. Sci. 179(13), 2232–2248 (2009).
[Crossref]

J. Opt. (2)

J. Zhu, H. Wei, Z. Xiaodong, and J. Guofan, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

K. Höschel and L. Vasudevan, “Genetic algorithms for lens design: a review,” J. Opt. 48(1), 134–144 (2019).
[Crossref]

Light Sci. Appl. (1)

T. Yang, G. F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light Sci. Appl. 6(10), e17081 (2017).
[Crossref] [PubMed]

Math. Model. Num. Opt. (1)

X.-S. Yang and D. Suash, “Engineering optimisation by cuckoo search,” Math. Model. Num. Opt. 1(4), 330–343 (2010).
[Crossref]

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref] [PubMed]

Neural Comput. Appl. (2)

S. Özyön, Y. Celal, and T. Hasan, “Incremental gravitational search algorithm for high-dimensional benchmark functions,” Neural Comput. Appl. 29, 1–25 (2018).

X.-S. Yang and D. Suash, “Cuckoo search: recent advances and applications,” Neural Comput. Appl. 24, 169–174 (2014).
[Crossref]

Opt. Commun. (1)

L. I. Jun, W. Huang, and F. Hongjie, “A novel method for finding the initial structure parameters of optical systems via a genetic algorithm,” Opt. Commun. 361, 28–35 (2016).
[Crossref]

Opt. Eng. (3)

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57(10), 101705 (2018).
[Crossref]

M. J. Kidger, “Use of the Levenberg-Marquardt (damped least-squares) optimization method in lens design,” Opt. Eng. 32, 1731–1740 (1993).
[Crossref]

M. I. Nikolic, B. Pablo, N. Bharathwaj, A. G. Dejan, L. Jayao, and M. Juan Carlos, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).

Opt. Express (1)

Optik (Stuttg.) (1)

X. Cheng, W. Yongtian, H. Qun, and I. Masaki, “Global and local optimization for optical systems,” Optik (Stuttg.) 117, 111–117 (2006).
[Crossref]

Proc. SPIE (4)

J. P. Rolland, K. Fuerschbach, G. E. Davis, and K. P. Thompson, “Pamplemousse: The optical design, fabrication, and assembly of a three-mirror freeform imaging telescope,” Proc. SPIE 9293, 92930L (2014).

L. Hazra and B. Saswatee, “Genetic algorithm in the structural design of Cooke triplet lenses,” Proc. SPIE 3737, 172–180 (1999).
[Crossref]

A. Broemel, H. Gross, D. Ochse, U. Lippmann, C. Ma, Y. Zhong, and M. Oleszko, “Performance comparison of polynomial representations for optimizing optical freeform systems,” Proc. SPIE 9626, 96260W (2015).

C. Menke, “Application of particle swarm optimization to the automatic design of optical systems,” Proc. SPIE 10690, 106901A (2018).
[Crossref]

Soft Comput. (1)

G.-G. Wang, D. Suash, A. H. Gandomi, Z. Zhaojun, and A. H. Alavi, “Chaotic cuckoo search,” Soft Comput. 20, 3349–3362 (2016).
[Crossref] [PubMed]

Struct. Multidis. Optim. (1)

A. Luersen, L. R. Marco, Rodolphe, and G. Frédéric, “A constrained, globalized, and bounded Nelder–Mead method for engineering optimization,” Struct. Multidis. Optim. 27, 43–54 (2004).

Other (13)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd edition: The Art of Scientific Computing (Cambridge University, 2007).

N. Hansen, F. Steffen, R. Raymond, and A. Auger, “Real-parameter black-box optimization benchmarking 2010: Experimental setup,” Research report RR-7215, INRIA, 2010.

COCO, (COmparing Continuous Optimisers) homepage, (INRIA, 2019), http://coco.gforge.inria.fr/ .

N. Hansen, F. Steffen, R. Raymond, and A. Auger, “Real-parameter black-box optimization benchmarking 2009: Noiseless functions definitions,” Research report RR-6829, INRIA, 2009.

N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009,” Proceedings of the 12th Annual Conference Companion on Genetic and Evolutionary Computation (2010). pp. 1689–1696.
[Crossref]

A. Hagg, A. Asteroth, and T. Bäck, “Prototype discovery using quality-diversity,” International Conference on Parallel Problem Solving from Nature (2018), pp. 500–511.
[Crossref]

Steel Bank Common Lisp (SBCL) homepage, (2019), http://www.sbcl.org/ .

“The CMA evolution strategy: A tutorial,” 2016, https://arxiv.org/pdf/1604.00772.pdf .

N. Hansen, “Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed,” Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers (2009), pp. 2389–2396.
[Crossref]

A. Auger and H. Nikolaus, “A restart CMA evolution strategy with increasing population size,” 2005 IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 1769–1776.
[Crossref]

R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science (1995), pp. 39–43.
[Crossref]

T. Zeugmann, P., Poupart and J. Kennedy, “Particle swarm optimization,” in Encyclopedia of Machine Learning, C. Sammut and G. I. Webb, ed. (Springer Science & Business Media, 2011).

D. Vasiljevic, Classical and Evolutionary Algorithms in the Optimization of Optical Systems (Springer Science & Business Media, 2012).

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Figures (11)

Fig. 1
Fig. 1 Typical flowchart for the work of an optical designer. The process highlighted in red is the one we focus on in the present paper.
Fig. 2
Fig. 2 Result of the implemented algorithms on the BBOB2009 test suite. Proportion of solved problems as a function of function evaluation budget. The threshold of the distance to the global minimum for a problem to be considered as solved is chosen arbitrarily to be Δfthresh = 1.
Fig. 3
Fig. 3 Result of the implemented algorithms on the BBOB2009 test suite. Proportion of solved problems as a function of threshold criterion Δfthresh. The function evaluation budget is 10^3 per problem dimension for every algorithm. The thick vertical line highlights the Δfthresh value taken in Fig. 2. The proportion of solved problems for each algorithm at this threshold matches the proportions obtained after 10^3 function evaluations per problem dimension in Fig. 2.
Fig. 4
Fig. 4 Layout of example System 1. Drawn using OS. For the purpose of the present simulation, the refractive index are taken invariant with respect to wavelength and n1 = n3 = n5 = 1.52 ; n2 = n4 = 1.64. The optimization variables are the thicknesses e1 through e11 and curvatures c1 through c10, there being 21 variables in total.
Fig. 5
Fig. 5 Layout of example System 2. Drawn using OS. S1 and S4 are plane refractive interfaces. S2 and S3 are reflective surfaces. The refractive index of the prism is n = 1.59. Surfaces S2 and S3 have 11 descriptive parameters which are used for the optimization: curvature of the base sphere and 10 XY polynomial coefficients. We drew blue lines connecting the various surfaces to give the readers an idea of what the prism might look like as a real object.
Fig. 6
Fig. 6 Result of the implemented algorithms and OS search methods on the MF of System 1. Number of successful runs (out of 100 starting systems) as a function of number of function evaluations per problem dimensionality (21 for System 1). The threshold of the MF value for a run to be considered successful is chosen arbitrarily to be Δfthresh = 10E-2. The thick black vertical line highlights the minimum evaluation budget for every run.
Fig. 7
Fig. 7 Result of the implemented algorithms and OS search methods on the MF of System 1. Number of successful runs (out of 100 starting systems) as a function of Δfthresh, the threshold in MF value below which a run is considered to be successful. The function evaluation budget is 5000 for every algorithm. The thick vertical line highlights the Δfthresh value taken in Fig. 6. The number of successful runs for each algorithm at this threshold matches the proportions obtained after 5000 function evaluations in Fig. 6.
Fig. 8
Fig. 8 Result of the implemented algorithms and OS search methods on the MF of System 2. Number of successful runs (out of 100 starting systems) as a function of number of function evaluations per problem dimensionality (22 for System 2). The threshold of the MF value for a run to be considered successful is chosen arbitrarily to be Δfthresh = 10E-2. The thick black vertical line highlights the minimum evaluation budget for every run, which is 5000.
Fig. 9
Fig. 9 Result of the implemented algorithms and OS search methods on the MF of System 2. Number of successful runs (out of 100 starting systems) as a function of Δfthresh, the threshold in MF value below which a run is considered to be successful. The function evaluation budget is 5000 for every algorithm. The thick vertical line highlights the Δfthresh value taken in Fig. 8. The number of successful runs for each algorithm at this threshold matches the proportions obtained after 5000 function evaluations in Fig. 8. The total number of runs for the SPX with restart is 292 and not shown on this diagram.
Fig. 10
Fig. 10 General flowchart for a global restart scheme strategy.
Fig. 11
Fig. 11 5 eyepieces (System 1) found using the restart scheme with PSO followed by the SPX. We use a distance metric of 1E-2 in Euclidian norm to remove duplicate results.

Tables (5)

Tables Icon

Table 1 Parameterization for each of the algorithms running on the BBOB2009 test suite.

Tables Icon

Table 2 Common specifications for the two optical systems shown in the present paper.

Tables Icon

Table 3 Specific design parameters for each optical system.

Tables Icon

Table 4 Reported errors in the validation of raytracing implementation and MF on the two example systems.

Tables Icon

Table 5 Parameterization for each of the algorithms running on the two optical case studies MF.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

v next =α× v last +β×( ran d 1 × D cog +ran d 2 × D soc )
G( t )= G 0 exp( αt n iter )
K best ( t )=round( n parts t( n parts 1 ) n iter )
x probe,Lévy = x +0.01σN( 0,1 ) | N( 0,1 ) | 0.66 N( 0,1 )( x x best )
x probe,walk = x +rand×( x shuff1 x shuff2 )
x new = x bary +α×( x bary x worst )
x new = x best +β×( x last x best )
2| y hi y lo | | y hi |+| y lo |+ε < f tol
MF= f=1 Nfields Spo t f +αPo s f Nfields
Spo t f = i=1 Nrays ( x i x ¯ ) 2 + ( y i y ¯ ) 2 Nrays
Po s f = ( x ¯ x ¯ trgt ) 2 + ( y ¯ y ¯ trgt ) 2