Statistics and Computing

, Volume 27, Issue 3, pp 845–863 | Cite as

Exact optimal experimental designs with constraints

  • Mercedes Esteban-Bravo
  • Agata Leszkiewicz
  • Jose M. Vidal-SanzEmail author


The experimental design literature has produced a wide range of algorithms optimizing estimator variance for linear models where the design-space is finite or a convex polytope. But these methods have problems handling nonlinear constraints or constraints over multiple treatments. This paper presents Newton-type algorithms to compute exact optimal designs in models with continuous and/or discrete regressors, where the set of feasible treatments is defined by nonlinear constraints. We carry out numerical comparisons with other state-of-art methods to show the performance of this approach.


Exact optimal experimental designs constrained designs Newton-type algorithms 



Research partly supported by MICINN (Spain) Grant ECO2011-30198 and ECO2015-67763-R. The authors would like to thank the Editor and two anonymous reviewers for their valuable comments. The names of the authors appear in alphabetical order.


  1. Addelman, S.: Symmetrical and asymmetrical fractional factorial plans. Technometrics 4(1), 47–58 (1962)MathSciNetCrossRefGoogle Scholar
  2. Atkinson, A.C., Donev, A.N., Tobias, R.D.: Optimum Experimental Designs, with SAS. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  3. Atwood, C.L.: Sequences converging to D-optimal designs of experiments. Ann. Stat. 1(2), 342–352 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Atwood, C.L.: Convergent design sequences, for sufficiently regular optimality criteria. Ann. Stat. 4(6), 1124–1138 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Borchers, B., Mitchell, J.E.: An improved branch and bound algorithm for mixed integer nonlinear programs. Comput. Oper. Res. 21(4), 359–367 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Borchers, B., Mitchell, J.E.: A computational comparison of branch and bound and outer approximation algorithms for 0–1 mixed integer nonlinear programs. Comput. Oper. Res. 24(8), 699–701 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9(4), 877–900 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Byrd, R.H., Schnabel, R.B., Shultz, G.A.: A trust region algorithm for nonlinearly constrained optimization. SIAM J. Numer. Anal. 24(5), 1152–1170 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chen, R.B., Chang, S.P., Wang, W., Tung, H.C., Wong, W.K.: Minimax optimal designs via particle swarm optimization methods. Stat. Comput. 25, 1–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cook, D., Fedorov, V.: Constrained optimization of experimental design. Statistics 26(2), 129–178 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cook, R.D., Nachtsheim, C.J.: A comparison of algorithms for constructing exact D-optimal designs. Technometrics 22(3), 315–324 (1980)CrossRefzbMATHGoogle Scholar
  12. Dennis Jr., J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM 16, 168–174 (1996)MathSciNetzbMATHGoogle Scholar
  13. Fedorov, V.V.: Theory of Optimal Experiments. Academic Press, New York (1972)Google Scholar
  14. Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, P.L., Wachter, A.: Global convergence of a trust-region sqp-filter algorithm for general nonlinear programming. SIAM J. Optim. 13(3), 635 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91(2), 239 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gondzio, J., Grothey, A.: Exploiting structure in parallel implementation of interior point methods for optimization. Comput. Manag. Sci. 6(2), 135–160 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Harman, R., Filová, L.: Computing efficient exact designs of experiments using integer quadratic programming. Comput. Stat. Data Anal. 71, 1159–1167 (2014)MathSciNetCrossRefGoogle Scholar
  18. Heinkenschloss, M., Ulbrich, M., Ulbrich, S.: Superlinear and quadratic convergence of affine-scaling interior-point newton methods for problems with simple bounds without strict complementarity assumption. Math. Program. 86(3), 615–635 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Johnson, M.E., Nachtsheim, C.J.: Some guidelines for constructing exact D-optimal designs on convex design spaces. Technometrics 25(3), 271–277 (1983)MathSciNetzbMATHGoogle Scholar
  20. Karlin, S., Studden, W.J.: Optimal experimental designs. Ann. Math. Stat. 37(4), 783–815 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kiefer, J.: Optimum experimental designs. J. R. Stat. Soc. Ser. B Stat. Methodol. 21(2), 272–319 (1959)MathSciNetzbMATHGoogle Scholar
  22. Kiefer, J., Wolfowitz, J.: Optimum designs in regression problems. Ann. Math. Stat. 30(2), 271–294 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kiefer, J., Wolfowitz, J.: The equivalence of two extremum problems. Can. J. Math. 12, 363–366 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kuhfeld, W.F., Tobias, R.D., Garratt, M.: Efficient experimental design with marketing research applications. J. Mark. Res. 31(4), 545–557 (1994)CrossRefGoogle Scholar
  25. Lin, C.D., Anderson-Cook, C.M., Hamada, M.S., Moore, L.M., Sitter, R.R.: Using genetic algorithms to design experiments: a review. Qual. Reliab. Eng. Int. 31(2), 155–167 (2015)CrossRefGoogle Scholar
  26. Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, New York (1999)zbMATHGoogle Scholar
  27. Mandal, A., Wong, W.K., Yu, Y.: Algorithmic searches for optimal designs. In: Dean, A., Morris, M., Stufken, J., Bingham, D. (eds.) Handbook of Design and Analysis of Experiments. Chapman and Hall/CRC, Boca Raton (2015)Google Scholar
  28. Mandal, A., Wu, C.J., Johnson, K.: Selc: sequential elimination of level combinations by means of modified genetic algorithms. Technometrics 48(2), 273–283 (2006)MathSciNetCrossRefGoogle Scholar
  29. Mitchell, T.J.: An algorithm for the construction of “D-optimal” experimental designs. Technometrics 16(2), 203–210 (1974)MathSciNetzbMATHGoogle Scholar
  30. Nguyen, N.K., Miller, A.J.: A review of some exchange algorithms for constructing discrete d-optimal designs. Comput. Stat. Data Anal. 14(4), 489–498 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2006)zbMATHGoogle Scholar
  32. Sagnol, G.: Computing optimal designs of multiresponse experiments reduces to second-order cone programming. J. Stat. Plan. Inference 141(5), 1684–1708 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. St. John, R.C., Draper, N.R.: D-optimality for regression designs: a review. Technometrics 17(1), 15–23 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Studden, W.J.: Elfving’s theorem and optimal designs for quadratic loss. Ann. Math. Stat. 42(5), 1613–1621 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Tseng, P.: Error bounds and superlinear convergence analysis of some newton-type methods in optimization. In: Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 445–462. Springer, New York (2000)CrossRefGoogle Scholar
  36. Ulbrich, M., Ulbrich, S., Vicente, L.N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Math. Program. 100(2), 379–410 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Vicente, L.N., Wright, S.J.: Local convergence of a primal-dual method for degenerate nonlinear programming. Comput. Optim. Appl. 22(3), 311–328 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Whittle, P.: Some general points in the theory of optimal experimental design. J. R. Stat. Soc. Ser. B Stat. Methodol. 35(1), 123–130 (1973)Google Scholar
  39. Woods, D.: Robust designs for binary data: applications of simulated annealing. J. Stat. Comput. Simul. 80(1), 29–41 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wright, S.J., Orban, D.: Properties of the log-barrier function on degenerate nonlinear programs. Math. Oper. Res. 27(3), 585–613 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Wynn, H.P.: The sequential generation of D-optimum experimental designs. Ann. Math. Stat. 41(5), 1655–1664 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Wynn, H.P.: Jack Kiefer’s contributions to experimental design. Ann. Stat. 12(2), 416–423 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Yu, Y.: D-optimal designs via a cocktail algorithm. Stat. Comput. 21(4), 475–481 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Mercedes Esteban-Bravo
    • 1
  • Agata Leszkiewicz
    • 2
  • Jose M. Vidal-Sanz
    • 1
    Email author
  1. 1.Department of Business AdministrationUniversidad Carlos III de MadridGetafeSpain
  2. 2.Center for Excellence in Brand and Customer ManagementRobinson College of Business, Georgia State UniversityAtlantaUSA

Personalised recommendations