Abstract

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the local minima form a network in which all nodes are connected. Each link between two neighboring minima contains a saddle point with a Morse index of 1. For a simple global optimization search (the symmetric Cooke triplet), the network of the corresponding set of local minima is presented.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Reiner and P. Pardalos, eds., Handbook of Global Optimization (Kluwer, Dordrecht, The Netherlands1995).
  2. A. E. W. Jones and G. W. Forbes, J. Global Optim. 6, 1 (1995).
    [CrossRef]
  3. T. G. Kuper and T. I. Harris, Proc. SPIE 1780, 14 (1992).
  4. K. E. Moore, Proc. SPIE 3780, 40 (1999).
    [CrossRef]
  5. M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
    [CrossRef]
  6. J. C. Hart, in Mathematical Visualization, H.-C. Hege and K. Polthier, eds. (Springer-Verlag, Berlin, 1998), p. 257.
    [CrossRef]
  7. E. van Driel, F. Bociort, and A. Serebriakov, “Topography of the merit function landscape in optical system design,” Proc. SPIE5249 (to be published).
  8. N. Mousseau and G. T. Barkema, Phys. Rev. E 57, 2419 (1998).
    [CrossRef]
  9. D. J. Wales, J. Chem. Phys. 101, 3750 (1994).

1999 (1)

K. E. Moore, Proc. SPIE 3780, 40 (1999).
[CrossRef]

1998 (1)

N. Mousseau and G. T. Barkema, Phys. Rev. E 57, 2419 (1998).
[CrossRef]

1995 (2)

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

A. E. W. Jones and G. W. Forbes, J. Global Optim. 6, 1 (1995).
[CrossRef]

1994 (1)

D. J. Wales, J. Chem. Phys. 101, 3750 (1994).

1992 (1)

T. G. Kuper and T. I. Harris, Proc. SPIE 1780, 14 (1992).

Barkema, G. T.

N. Mousseau and G. T. Barkema, Phys. Rev. E 57, 2419 (1998).
[CrossRef]

Bociort, F.

E. van Driel, F. Bociort, and A. Serebriakov, “Topography of the merit function landscape in optical system design,” Proc. SPIE5249 (to be published).

Forbes, G. W.

A. E. W. Jones and G. W. Forbes, J. Global Optim. 6, 1 (1995).
[CrossRef]

Harris, T. I.

T. G. Kuper and T. I. Harris, Proc. SPIE 1780, 14 (1992).

Hart, J. C.

J. C. Hart, in Mathematical Visualization, H.-C. Hege and K. Polthier, eds. (Springer-Verlag, Berlin, 1998), p. 257.
[CrossRef]

Hiraga, K.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Ishikawa, J.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Isshiki, M.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Jones, A. E. W.

A. E. W. Jones and G. W. Forbes, J. Global Optim. 6, 1 (1995).
[CrossRef]

Kuper, T. G.

T. G. Kuper and T. I. Harris, Proc. SPIE 1780, 14 (1992).

Moore, K. E.

K. E. Moore, Proc. SPIE 3780, 40 (1999).
[CrossRef]

Mousseau, N.

N. Mousseau and G. T. Barkema, Phys. Rev. E 57, 2419 (1998).
[CrossRef]

Nakadate, S.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Ono, H.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Serebriakov, A.

E. van Driel, F. Bociort, and A. Serebriakov, “Topography of the merit function landscape in optical system design,” Proc. SPIE5249 (to be published).

van Driel, E.

E. van Driel, F. Bociort, and A. Serebriakov, “Topography of the merit function landscape in optical system design,” Proc. SPIE5249 (to be published).

Wales, D. J.

D. J. Wales, J. Chem. Phys. 101, 3750 (1994).

J. Chem. Phys. (1)

D. J. Wales, J. Chem. Phys. 101, 3750 (1994).

J. Global Optim. (1)

A. E. W. Jones and G. W. Forbes, J. Global Optim. 6, 1 (1995).
[CrossRef]

Opt. Rev. (1)

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, Opt. Rev. 6, 463 (1995).
[CrossRef]

Phys. Rev. E (1)

N. Mousseau and G. T. Barkema, Phys. Rev. E 57, 2419 (1998).
[CrossRef]

Proc. SPIE (2)

T. G. Kuper and T. I. Harris, Proc. SPIE 1780, 14 (1992).

K. E. Moore, Proc. SPIE 3780, 40 (1999).
[CrossRef]

Other (3)

J. C. Hart, in Mathematical Visualization, H.-C. Hege and K. Polthier, eds. (Springer-Verlag, Berlin, 1998), p. 257.
[CrossRef]

E. van Driel, F. Bociort, and A. Serebriakov, “Topography of the merit function landscape in optical system design,” Proc. SPIE5249 (to be published).

H. Reiner and P. Pardalos, eds., Handbook of Global Optimization (Kluwer, Dordrecht, The Netherlands1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Two connected minima and a saddle point on the link.

Fig. 2
Fig. 2

Three connected minima. The links (dotted curves) are the paths of local optimization started close to the saddle point on both sides along the downward direction. The EMS (continuous curves) have been obtained from a two-dimensional cut through the five-dimensional merit function landscape of a Cooke triplet global search.

Fig. 3
Fig. 3

Network of the global search corresponding to a symmetric Cooke triplet.

Metrics