Skip to main content
SpringerLink
Log in

Nonsmooth optimization through Mesh Adaptive Direct Search and Variable Neighborhood Search

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

Abstract

This paper proposes a way to combine the Mesh Adaptive Direct Search (MADS) algorithm, which extends the Generalized Pattern Search (GPS) algorithm, with the Variable Neighborhood Search (VNS) metaheuristic, for nonsmooth constrained optimization. The resulting algorithm retains the convergence properties of MADS, and allows the far reaching exploration features of VNS to move away from local solutions. The paper also proposes a generic way to use surrogate functions in the VNS search. Numerical results illustrate advantages and limitations of this method.

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

Similar content being viewed by others

References

  1. Abramson M.A. (2004). Mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm. Optim. Eng. 5(2): 157–177

    Article  Google Scholar 

  2. Abramson M.A. and Audet C. (2006). Second-order convergence of mesh-adaptive direct search. SIAM J. Optim. 17(2): 606–619

    Article  Google Scholar 

  3. Alberto P., Nogueira F., Rocha U. and Vicente L.N. (2004). Pattern search methods for user-provided points: application to molecular geometry problems. SIAM J. Optim. 14(4): 1216–1236

    Article  Google Scholar 

  4. Audet C., Brimberg J., Hansen P., Le Digabel S. and Mladenović N. (2004). Pooling problem: alternate formulations and solution methods. Manage. Sci. 50(6): 761–776

    Article  Google Scholar 

  5. Audet, C., Couture, G., Dennis, J.E. Jr.: NOMAD Project (LTMADS Package). Software available at www.gerad.ca/nomad

  6. Audet C. and Dennis J.E. (2000). Pattern search algorithms for mixed variable programming. SIAM J. Optim. 11(3): 573–594

    Article  Google Scholar 

  7. Audet C. and Dennis J.E. (2003). Analysis of generalized pattern searches. SIAM J. Optim. 13(3): 889–903

    Article  Google Scholar 

  8. Audet C. and Dennis J.E. (2004). A pattern search filter method for nonlinear programming without derivatives. SIAM J. Optim. 14(4): 980–1010

    Article  Google Scholar 

  9. Audet C. and Dennis J.E. (2006). Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(1): 188–217

    Article  Google Scholar 

  10. Audet C. and Orban D. (2006). Finding optimal algorithmic parameters using the mesh adaptive direct search algorithm. SIAM J. Optim. 17(3): 642–664

    Article  Google Scholar 

  11. Béchard, V., Audet, C., Chaouki, J.: Robust optimization of chemical processes using a MADS algorithm. Technical reportG–2005–16, Les Cahiers du GERAD, Montréal (2005)

  12. Booker, A.J., Dennis, J.E. Jr., Frank, P.D., Moore, D.W., Serafini, D.B.: Managing surrogate objectives to optimize a helicopter rotor design—further experiments. AIAA Paper 1998–4717, Presented at the 8th AIAA/ISSMO symposium on multidisciplinary analysis and optimization, St. Louis (1998)

  13. Booker A.J., Frank P.D., Serafini D.B., Torczon V. and Dennis J.E. (1998). Optimization using surrogate objectives on a helicopter test example. In: Borggaard, J., Burns, J., Cliff, E., and Schreck, S. (eds) Optimal Des. Control, Progress in Systems and Control Theory, pp 49–58. Birkhäuser, Cambridge

    Google Scholar 

  14. Booker A.J., Frank P.D., Serafini D.B., Torczon V., Trosset M.W. and Dennis J.E. (1999). A rigorous framework for optimization of expensive functions by surrogates. Struct. Optim. 17(1): 1–13

    Article  Google Scholar 

  15. Bowden, R.O., Hall, J.D.: Simulation optimization research and development. Simulation conference, 1693–1698, Presented at the 1998 winter simulation conference (1998)

  16. Brimberg, J., Mladenović, N.: A variable neighbourhood algorithm for solving the continuous location-allocation problem. In: Hamacher, D. (ed.) S tud. Location Anal. vol. 10, pp. 1–12, Athens, Greece (1996)

  17. Caporossi G. and Hansen P. (2000). Variable neighborhood search for extremal graphs 1: the AutoGraphiX system. Discrete Math. 212: 29–44

    Article  Google Scholar 

  18. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reissued in 1990 by SIAM Publications, Philadelphia, as vol. 5 in the series Classics in Applied Mathematics

  19. Douglas J.M. (1988). Conceptual Design of Chemical Processes. McGraw-Hill, New York

    Google Scholar 

  20. Drazić, M., Lavor, C., Maculan, N., Mladenović, N.: A Continuous VNS Heuristic for Finding the Tridimensional Structure of a Molecule. Technical Report G–2004–22, Les Cahiers du GERAD, Montréal (2004)

  21. Finkel, D.E., Kelley, C.T.: Convergence analysis of the direct algorithm. SIAM J. Optim. 2004 (to appear)

  22. Fletcher R. and Leyffer S. (2002). Nonlinear programming without a penalty function. Math. Program. Series A 91: 239–269

    Article  Google Scholar 

  23. Fowler K.R., Kelley C.T., Miller C.T., Kees C.E., Darwin R.W., Reese J.P., Farthing M.W. and Reed M.S.C. (2004). Solution of a well-field design problem with implicit filtering. Optim. Eng. 5(2): 207–234

    Article  Google Scholar 

  24. Hansen P. and Mladenović N. (2001). J-MEANS: a new local search heuristic for minimum sum of squares clustering. Pattern Recogn. 34(2): 405–413

    Article  Google Scholar 

  25. Hansen P. and Mladenović N. (2001). Variable neighborhood search: principles and applications. Eur. J. Oper. Res. 130(3): 449–467

    Article  Google Scholar 

  26. Hansen P., Mladenović N. and Perez-Britos D. (2001). Variable neighborhood decomposition search. J. Heuristics 7(4): 335–350

    Article  Google Scholar 

  27. Hansen P., Mladenović N. and Urosevic D. (2004). Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145(1): 117–125

    Article  Google Scholar 

  28. Hayes R.E., Bertrand F.H., Audet C. and Kolaczkowski S.T. (2003). Catalytic combustion kinetics: Using a direct search algorithm to evaluate kinetic parameters from light-off curves. Can. J. Chem. Eng. 81(6): 1192–1199

    Article  Google Scholar 

  29. Himmelblau D.M., Edgar T.F. and Lasdon L.S. (2003). Optimization of Chemical Processes, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  30. Jones D.R., Perttunen C.D. and Stuckman B.E. (1993). Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1): 157–181

    Article  Google Scholar 

  31. Kokkolaras M., Audet C. and Dennis J.E. (2001). Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system. Optim. Eng. 2(1): 5–29

    Article  Google Scholar 

  32. Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45(3):385–482 (electronic) (2003)

    Google Scholar 

  33. Lewis R.M., Torczon V. and Trosset M.W. (2000). Direct search methods: Then and now. J. Comput. Appl. Math. 124(1–2): 191–207

    Article  Google Scholar 

  34. Lophaven, S., Nielsen, H., Sondergaard, J.: DACE—A Matlab Kriging toolbox, version 2.0. Technical report IMM-REP-2002-12, Informatics and mathematical modelling, Technical University of Denmark (2002)

  35. Marsden A.L., Wang M., Moin P. and Dennis J.E. (2004). Optimal aeroacoustic shape design using the surrogate management framework. Optim. Eng. 5(2): 235–262

    Article  Google Scholar 

  36. Mladenović N. and Hansen P. (1997). Variable neighborhood search. Comput. Oper. Res. 24(11): 1097–1100

    Article  Google Scholar 

  37. Perez, R., Liu, H.H.T., Behdinan, K.: Evaluation of multidisciplinary optimization approaches for aircraft conceptual design. In: AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY, September (2004)

  38. Seader J.D., Sieder W.D. and Lewin D.R. (1999). Process Design Principles: Synthesis, Analysis and Evaluation. John Wiley and Sons Inc., New York

    Google Scholar 

  39. Serafini, D.B.: A framework for managing models in nonlinear optimization of computationally expensive functions. Ph.D. thesis, Department of Mathematical Sciences, Rice University, Houston, Texas (1998)

  40. Snyder J.D. and Subramaniam B. (1994). A novel reverse flow strategy for ethylbenzene dehydrogenation in a packed-bed reactor. Chem. Eng. Sci. 49: 5585–5601

    Article  Google Scholar 

  41. Sobieszczanski-Sobieski, J., Agte, J.S., Sandusky, R.R. Jr.: Bi-level integrated system synthesis (BLISS). Technical Report NASA/TM-1998-208715, NASA, Langley Research Center, August 1998

  42. Stein M. (1987). Large sample properties of simulations using latin hypercube sampling. Technometrics 29(2): 143–151

    Article  Google Scholar 

  43. Tang B. (1993). Orthogonal array-based latin hypercubes. J. Am. Stat. Assoc. 88(424): 1392–1397

    Article  Google Scholar 

  44. Timmerhaus K.D., Peters M.S. and West R.E. (2003). Plant Design and Economics for Chemical Engineers, 5th edn. McGraw-Hill, New York

    Google Scholar 

  45. Torczon V. (1997). On the convergence of pattern search algorithms. SIAM J. Optim. 7(1): 1–25

    Article  Google Scholar 

  46. Trefethen, N.: The hundred dollar hundred digit challenge. SIAM News 35(1), January–February 2002. www.siam.org/news/news.php?id=388

  47. Vaz A.I.F. and Vicente L.N. (2007). A particle swarm pattern search method for bound constrained global optimization. J. Global Optim. 39(2): 197–219

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques et de Génie Industriel, Ecole Polytechnique de Montréal and GERAD, C.P. 6079, Succ. Centre-ville, Montreal, QC, Canada, H3C 3A7

    Charles Audet, Vincent Béchard & Sébastien Le Digabel

Authors
  1. Charles Audet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincent Béchard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sébastien Le Digabel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sébastien Le Digabel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Audet, C., Béchard, V. & Digabel, S.L. Nonsmooth optimization through Mesh Adaptive Direct Search and Variable Neighborhood Search. J Glob Optim 41, 299–318 (2008). https://doi.org/10.1007/s10898-007-9234-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-007-9234-1

Keywords

Search

Navigation