## Mutation operators in lens system optimization to jump out of local minima |

Optics Express, Vol. 18, Issue 20, pp. 20876-20886 (2010)

http://dx.doi.org/10.1364/OE.18.020876

Acrobat PDF (1607 KB)

### Abstract

In lens system design, Damped Least Squares method is a traditionally used local search method. In this paper we apply mutation operators to control damping factor in Damped Least Squares. We study the mutation behavior in this method when the algorithm confronts with local minima. The proposed method can go beyond local minima by taking mutation operators to control damping factor. The proposed method was successfully applied to design problems. The result indicates that the mutation operators provide an effective and rapid way to jump out of poor local minima.

© 2010 OSA

## 1. Introduction

2. M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. **2**(6), 463–470 (1995). [CrossRef]

8. L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express **18**(12), 13285–13300 (2010). [CrossRef] [PubMed]

10. M. van Turnhout and F. Bociort, “Chaotic behavior in an algorithm to escape from poor local minima in lens design,” Opt. Express **17**(8), 6436–6450 (2009). [CrossRef] [PubMed]

## 2. Theory

11. J. Meiron, “Damped Least-Squares method for automatic lens design,” J. Opt. Soc. Am. **55**(9), 1105–1107 (1965). [CrossRef]

*∆X*to change the configuration parameter vector

*X*of a lens system as follows:

*M*) and the variable space (dimension

*N*).

*A*(

*M*×

*N*) is derivative matrix of an optimization problem.

*I*(

*N*×

*N*) is the identity matrix.

*F*is the vector of operand values to be reduced to zero or certain target values.

_{0}*X*is the configuration parameter vector before iteration.

_{0}*P*is the damping factor. Usually

*P*is a proper positive real number. The choice of

*P*determines the effectiveness of the algorithm. For more details, see Ref [11

11. J. Meiron, “Damped Least-Squares method for automatic lens design,” J. Opt. Soc. Am. **55**(9), 1105–1107 (1965). [CrossRef]

*P*when the solution is trapped in local minima. From the Eq. (1), we know that if

*∆X*is to become small and the algorithm is to converge due to basins of attraction [12

12. H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science **271**(5254), 1376–1380 (1996). [CrossRef]

*P*is the eigenvalue randomly chosen from all the eigenvalues of the matrix

_{k}*P*to -

*P*,

_{k}*∆X*becomes infinite and out of control. The obtained configuration parameter vector

*X*is usually useless. However, in this paper, we proposed a mutation operator based on Self-Adaptive Gaussian mutation and Decreasing-Based Gaussian mutation [13,14], which are traditionally used in evolutionary algorithm to keep diversity of the population, to control the damping factor

*P*as follows: where

*N(0,1)*is the standard normal distribution,

*N*is a new value with distribution

_{i}(0,1)*N(0,1)*that must be regenerated for each index

*i*.

*i*presents number of iteration.

*n*is a positive integer.

*Q*is a constant whose value is between 0 and 1. In the proposed method, the mutated

*P*as shown in Eq. (5) searches around a center -

*P*, and is neither too near to nor too far from -

_{k}*P*due to the proposed mutation operator. In this way, the matrix

_{k}*P*can produce a large amount of different

*∆X*resulting in mutated solution

*X*. The

*∆X*resulting from the mutated

*P*is larger than that in trapped local minimum. In this way, the algorithm is no longer to stay in the local minimum. On the contrary, the algorithm with mutated

*P*is to search more extensive variable space beyond the basins so that it can help the trapped solution escape from local minima.

*P*as damping factor, mutation period. The mutation period is usually applied to search the variable space until it produce solutions better than the trapped solution. The mutated

*P*is neither too near to -

*P*, which makes

_{k}*P*, which results in no obvious mutation taking place.

_{k}## 3. Mutation behavior in Damped Least Squares method

*r*) and thicknesses (

_{1}, r_{2}, r_{3}, r_{4}*d*) as shown in Fig. 1 . The stop is placed at the first surface. The configuration parameter vector

_{1}, d_{2}, d_{3}, d_{4}*X*is (

*r*).

_{1}, r_{2}, r_{3}, r_{4}, d_{1}, d_{2}, d_{3}, d_{4}*P*as shown in Eq. (5). The algorithm continues to iterate. This procedure is mutation period as we mention in Section 2. Then the mutation period is followed by the former period (using conventional damping factor). These two periods alternate until the final satisfactory result is produced. In the following experiments,

*Q*is set to 0.7.

*n*to 3. Figure 2 shows the value of merit function for each iteration in the optimization procedure. We allow the algorithm to iterate for 1200 times. The vertical axis is the values of merit function, and the horizontal axis is the times of iteration. During the mutation period (600-700, 800-900, 1000-1100), we reset

*P*according to Eq. (5). In these periods, we find that the merit function varies randomly shown in Fig. 2 (a). This is because the damping factor is controlled by mutation operators to change randomly, which results in the change of

*X*to search variable space to jump out of the local minimum. In Fig. 2 (b), the mutation has produced many solutions that are better than the trapped solution. Then we chose the best one (the one with the smallest value of merit function) as the initial solution for the next search procedure after 700 iterations. Figure 2 (c) and Fig. 2 (d) also show the same kind of phenomenon.

*n*to 13. Figure 3 shows the value of merit function for each iteration in the optimization procedure. In this trial,

*n*becomes larger than that in the first one. We could see that this algorithm iterates less than 1050, but it has produced a satisfactory result which is better than the target. Thus the algorithm ends ahead of 1200 times.

*n*to 23. The same phenomenon happens as shown in Fig. 4 . We can see that in the first mutation period the value of merit function decreases sharply so as it needn’t more iteration times to produce a satisfactory result.

*P*is neither too near to nor too far from -

*P*. Therefore, in the fourth trial, we make mutated

_{k}*P*very near to -

*P*as follow:

_{k}*P*in Eq. (8) searches the space only near to -

*P*(very near with high probability).

_{k}*n*is set to 1.

*P*is too near to -

*P*, and

_{k}*∆X*becomes infinite and out of control. Thus result is very bad.

*P*far from -

*P*as follow: where

_{k}*P*is not eigenvalue and is set to 0.2. The value is similar to the value of the common damping factor so as to avoid singular matrix, so the center

_{0}*P*is quite far from -

_{0}*P*. We could hardly find the mutation phenomenon during iteration process in Fig. 6 .

_{k}*P*controlled by mutation operators should be set neither too near to nor too far from -

*P*. In this paper, we propose a mutation operator shown in Eq. (5). It works fairly well in the optimization.

_{k}## 4. Design examples and result discussions

15. D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE **1354**, 548–554 (1990). [CrossRef]

*f/3*, 100 mm effective focal length lens of BK7 glass, illuminated by helium

*d*wavelength. The object is at infinity, the object field covers 30° full field (15° semi-field angle) and the image field is flat. Constraints on the construction include: only spherical surfaces, no aspheric, GRIN elements, Fresnel lenses, binary elements, holographic optical elements, etc. The minimum glass thickness is 2 mm, but there is no upper limit on the size of the lens. The distortion must be less than 1% and there should be no vignetting. The last is intended to assure that vignetting could not be used to improve the edge performance on the lens. No requirement is put on the location of the stop of the system. The merit function consists of the average of the RMS blur spot for three fields: on-axis, 10.5°, and 15°, weighted equally.

*P*,

_{dist}*P*and

_{img}*P*are different penalties computed from a set of physical constraints, as presented in Table 1 . The initial value of

_{vign}*P*,

_{dist}*P*and

_{img}*P*is 0. The penalty value is used to make sure the merit function value of unfeasible designs is always worse than that of feasible one. The RMS blur spot size is computed from the variance of the position at the image plane of different exact rays with the same entrance angle.

_{vign}*P*is changed by mutation operators. This operation makes the algorithm search more extensive variable space, which helps the trapped solution jump out of the local minimum successfully.

## 5. Conclusion

## Acknowledgments

## References and links

1. | K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. |

2. | M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. |

3. | D. Vasiljevic, “Classical and evolutionary algorithms in the optimization of optical systems,” (Wiley, 2001). |

4. | V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. |

5. | C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. |

6. | I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. |

7. | L. Li, Q. H. Wang, D. H. Li, and H. R. Peng, “Jump method for optical thin film design,” Opt. Express |

8. | L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express |

9. | D. Shafer, “Global optimization in optical design,” Comput. Phys. |

10. | M. van Turnhout and F. Bociort, “Chaotic behavior in an algorithm to escape from poor local minima in lens design,” Opt. Express |

11. | J. Meiron, “Damped Least-Squares method for automatic lens design,” J. Opt. Soc. Am. |

12. | H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science |

13. | H. P. Schwefel, “Numerical optimization of computer models,” (Wiley, 1981). |

14. | J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. |

15. | D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE |

**OCIS Codes**

(220.2740) Optical design and fabrication : Geometric optical design

(220.3620) Optical design and fabrication : Lens system design

(080.1753) Geometric optics : Computation methods

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: June 8, 2010

Revised Manuscript: August 13, 2010

Manuscript Accepted: September 15, 2010

Published: September 17, 2010

**Citation**

Lei Li, Qiong-Hua Wang, Xiao-Qing Xu, and Da-Hai Li, "Mutation operators in lens system optimization to jump out of local minima," Opt. Express **18**, 20876-20886 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20876

Sort: Year | Journal | Reset

### References

- K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).
- M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995). [CrossRef]
- D. Vasiljevic, “Classical and evolutionary algorithms in the optimization of optical systems,” (Wiley, 2001).
- V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41(30), 6514–6520 (2002). [CrossRef] [PubMed]
- C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008). [CrossRef]
- I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000). [CrossRef]
- L. Li, Q. H. Wang, D. H. Li, and H. R. Peng, “Jump method for optical thin film design,” Opt. Express 17(19), 16920–16926 (2009). [CrossRef] [PubMed]
- L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express 18(12), 13285–13300 (2010). [CrossRef] [PubMed]
- D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
- M. van Turnhout and F. Bociort, “Chaotic behavior in an algorithm to escape from poor local minima in lens design,” Opt. Express 17(8), 6436–6450 (2009). [CrossRef] [PubMed]
- J. Meiron, “Damped Least-Squares method for automatic lens design,” J. Opt. Soc. Am. 55(9), 1105–1107 (1965). [CrossRef]
- H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996). [CrossRef]
- H. P. Schwefel, “Numerical optimization of computer models,” (Wiley, 1981).
- J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).
- D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE 1354, 548–554 (1990). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.