Skip to main content
SpringerLink
Log in

Geodesic and contour optimization using conformal mapping

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We propose a novel optimization algorithm for differentiable functions utilizing geodesics and contours under conformal mapping. The algorithm can locate multiple optima by first following a geodesic curve to a local optimum then traveling to the next search area by following a contour curve. Alongside we implement a jumping mechanism which we call shadow casting to help geodesics jump to locations closer to the global optimum. To improve the efficiency, local search methods such as the Newton–Raphson algorithm are also employed. For functions with many optima or when the global optimum is very close to a local one, numerical analyses have shown that the resulting algorithm, SGEO-QN, can outperform recent derivative-free DIRECT variants in number of function/gradient evaluations. The results also indicate that under certain conditions, number of function/gradient evaluations for SGEO-QN scales nearly linearly with increasing dimensionality. Lastly, SGEO-QN appears to be less affected by rotational transforms of the objective functions than the variants of DIRECT compared.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. We use a shorthand for partial derivatives, \(\partial _i \equiv \frac{\partial }{\partial x^i}\).

  2. Here, \(\mathbf {v}\) and \(\nabla f\) represent the tangent and gradient vectors in Euclidean space.

  3. Note that \( \sum _t f(\mathbf {x}_t) \ne 1\) and the vector \(\sum _t \widehat{f}(\mathbf {x}_t) (\mathbf {x}_t - \mathbf {A})\) will be approximately parallel to \((\mathbf {x}_t - \mathbf {A})\) in Eq. (6).

  4. Note that \(\mathbf {x}^*_{\gamma } \in L\).

  5. This is similar to the monotonic basin hopping algorithm [27] but we use a random walk instead of uniform sampling in a D-sphere.

  6. We implement quasi-Newton (QN) local searches on the geodesics \(\gamma \) and the line L. Hence we call the resulting algorithm SGEO-QN.

  7. A recent publication tackling the problem of comparing deterministic and stochastic optimization methods is discussed in [34].

  8. The only quantity that depends on the coordinate is \(t_C\) in Eq. (9), where a different choice of basis would result in different values in the components.

  9. The composition functions have the form \(f(x) = \sum _i \omega _i g_i(x)\), where i denotes the i-th basic function \(g_i(x)\) and its weight \(\omega _i \in [0,1]\), such that \(\sum _i \omega _i = 1\).

References

  1. Diener, I.: Handbook of Global Optimization, Vol. 2 of the Series Nonconvex Optimization and its Applications, pp. 649–668 (1995)

  2. Floudas, C.A., Pardalos, P. M.: Encyclopedia of Optimization, Globally Convergent Homotopy Methods, pp. 1272–1277

  3. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeros of maps: homotopy methods that are constructive with probability one. Math. Comput. 32, 887899 (1978)

    Article  MATH  Google Scholar 

  4. Watson, L.T.: Globally convergent homotopy algorithms for nonlinear systems of equations. Nonlinear Dyn. 1, 143–191 (1990)

    Article  Google Scholar 

  5. Dunlavy, D.M., O’Leary, D.P.: Homotopy Optimization Methods for Global Optimization, Sandia National Laboratories, Report SAND 2005–7495 (2005)

  6. Botsaris, C.A.: Constrained optimization along geodesics. J. Math. Anal. Appl. 79(2), 295–306 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Smith, S.T.: Geometric Optimization Methods for Adaptive Filtering. Harvard University, Cambridge (1993)

    Google Scholar 

  8. Smith, S.T.: Optimization techniques on Riemannian manifolds. Fields Inst. Commun. 3(3), 113–135 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Rapcsák, T.: Convex Programming on Riemannian Manifolds, Lecture Notes in Control and Information Sciences, vol. 84, pp. 733–740. Springer, Berlin (1986)

  10. Rapcsák, T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69(1), 169–183 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rapcsák, T.: Geodesic convexity on \(\mathbb{R}^n_+\). Optimization 37(4), 341–355 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rapcsák, T.: Local convexity on smooth manifolds. J. Optim. Theory Appl. 127(1), 165–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Perttunen, C.D., Jones, D.R., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157181 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Liuzzi, G., Lucidi, S., Piccialli, V.: A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems. Comput. Optim. Appl. 45(2), 353–375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liuzzi, G., Lucidi, S., Piccialli, V.: Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization. Comput. Optim. Appl. (2015). doi:10.1007/s10589-015-9741-9

  16. Kvasov, D.E., Sergeyev, Y.D.: A univariate global search working with a set of Lipschitz constants for the first derivative. Optim. Lett. 3(2), 303–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kvasov, D.E., Sergeyev, Y.D.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236(16), 4042–4054 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Butz, A.R.: Space-filling curves and mathematical programming. Inf. Control 12(4), 313–330 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lera, D., Sergeyev, Y.D.: Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Hölder constants. Commun. Nonlinear Sci. Numer. Simul. 23(1–3), 328–342 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, Berlin (2013). doi:10.1007/978-1-4614-8042-6

    Book  MATH  Google Scholar 

  21. Schoen, F.: Two-Phase Methods for Global Optimization, Handbook of Global Optimization, Vol. 62 of the Series Nonconvex Optimization and its Applications, pp. 151–177 (2002)

  22. Sergeyev, Y., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16(3), 910937 (2006). doi:10.1137/040621132

    Article  MathSciNet  MATH  Google Scholar 

  23. Boothby, M.W.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Springer, Berlin (2003)

    MATH  Google Scholar 

  24. Lee, M.J.: Introduction to Topological Manifolds. Springer, Berlin (2010)

    Google Scholar 

  25. Foster, J.: A Short Course in General Relativity. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  26. Hawking, S., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  27. Wales, D.J., Doye, J.P.K.: Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones Clusters containing up to 110 atoms. Phys. Chem. A 101, 5111 (1997)

    Article  Google Scholar 

  28. Gaviano, M., Kvasov, D.E., Lera, D., Sergeyev, Y.D.: Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29(4), 469–480 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Addis, B., Locatelli, M.: A new class of test functions for global optimization. J. Glob. Optim. 38(3), 479–501 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zilinskas, A.: A class of test functions for global optimization. J. Glob. Optim. 5(2), 195–199 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schoen, F.: A wide class of test functions for global optimization. J. Glob. Optim. 3, 133–138 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hedar, A.R.: Test Problems for Unconstrained Optimization, http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO_files/Page364.htm

  33. Liang, J.J., Qu, B.Y., Suganthan, P.N., Hernndaz-Daz, A.G.: Problem Definitions and Evaluation Criteria for the CEC 2013 Special Session and Competition on Real-parameter Optimization, Technical Report 201212. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China (2013)

  34. Sergeyev, Y., Kvasov, D.E., Mukhametzhanov, M.S.: Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms. Math. Comput. Simul. (2016). doi:10.1016/j.matcom.2016.05.006

  35. Jamil, M., Yang, X.S.: A literature survey of benchmark functions for global optimisation problems. Int. J. Math. Model. Numer. Optim. 4(2), 150–194 (2013)

    MATH  Google Scholar 

Download references

Acknowledgments

We thank the anonymous reviewers for extremely useful feedbacks in assisting the writing of this manuscript.

Author information

Authors and Affiliations

  1. Department of Computer Science, York University, 4700 Keele Street, Toronto, M3J 1P3, Canada

    Ricky Fok & Aijun An

  2. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, M3J 1P3, Canada

    Xiaogong Wang

Authors
  1. Ricky Fok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aijun An
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaogong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricky Fok.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Funding

This research is supported by the Discovery Grants and Discovery Accelerator Supplement from Natural Sciences and Engineering Research Council of Canada (NSERC).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fok, R., An, A. & Wang, X. Geodesic and contour optimization using conformal mapping. J Glob Optim 69, 23–44 (2017). https://doi.org/10.1007/s10898-016-0467-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-016-0467-8

Keywords

Search

Navigation