Optimization Letters

, Volume 11, Issue 5, pp 895–913 | Cite as

ARGONAUT: AlgoRithms for Global Optimization of coNstrAined grey-box compUTational problems

  • Fani Boukouvala
  • Christodoulos A. FloudasEmail author
Original Paper


The algorithmic framework ARGONAUT is presented for the global optimization of general constrained grey-box problems. ARGONAUT incorporates variable selection, bounds tightening and constrained sampling techniques, in order to develop accurate surrogate representations of unknown equations, which are globally optimized. ARGONAUT is tested on a large set of test problems for constrained global optimization with a large number of input variables and constraints. The performance of the presented framework is compared to that of existing techniques for constrained derivative-free optimization.


Grey-box optimization Surrogate modeling Variable selection Derivative-free optimization General constraints Nonlinear programming 



The authors acknowledge financial support from the National Science Foundation (CBET-0827907, CBET-1263165).

Supplementary material

11590_2016_1028_MOESM1_ESM.doc (340 kb)
Supplementary material 1 (doc 340 KB)


  1. 1.
    Audet, C., Bechard, V., Chaouki, J.: Spent potliner treatment process optimization using a MADS algorithm. Optim. Eng. 9(2), 143–160 (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bartholomew-Biggs, M.C., Parkhurst, S.C., Wilson, S.P.: Using DIRECT to solve an aircraft routing problem. Comput. Optim. Appl. 21(3), 311–323 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boukouvala, F., Ierapetritou, M.G.: Surrogate-based optimization of expensive flowsheet modeling for continuous pharmaceutical manufacturing. J. Pharm. Innov. 8(2), 131–145 (2013)CrossRefGoogle Scholar
  4. 4.
    Caballero, J.A., Grossmann, I.E.: An algorithm for the use of surrogate models in modular flowsheet optimization. Aiche J. 54(10), 2633–2650 (2008)CrossRefGoogle Scholar
  5. 5.
    Egea, J.A., Rodriguez-Fernandez, M., Banga, J.R., Marti, R.: Scatter search for chemical and bio-process optimization. J. Global Optim. 37(3), 481–503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fahmi, I., Cremaschi, S.: Process synthesis of biodiesel production plant using artificial neural networks as the surrogate models. Comput. Chem. Eng. 46, 105–123 (2012)CrossRefGoogle Scholar
  7. 7.
    Fowler, K.R., Reese, J.P., Kees, C.E., Dennis Jr., J.E., Kelley, C.T., Miller, C.T., Audet, C., Booker, A.J., Couture, G., Darwin, R.W., Farthing, M.W., Finkel, D.E., Gablonsky, J.M., Gray, G., Kolda, T.G.: Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems. Adv. Water Resour. 31(5), 743–757 (2008)CrossRefGoogle Scholar
  8. 8.
    Graciano, J.E.A., Roux, G.A.C.L.: Improvements in surrogate models for process synthesis. Application to water network system design. Comput. Chem. Eng. 59, 197–210 (2013)CrossRefGoogle Scholar
  9. 9.
    Hemker, T., Fowler, K., Farthing, M., Stryk, O.: A mixed-integer simulation-based optimization approach with surrogate functions in water resources management. Optim. Eng. 9(4), 341–360 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Henao, C.A., Maravelias, C.T.: Surrogate-based superstructure optimization framework. AIChE J. 57(5), 1216–1232 (2011)CrossRefGoogle Scholar
  11. 11.
    Kleijnen, J.P.C., van Beers, W., van Nieuwenhuyse, I.: Constrained optimization in expensive simulation: novel approach. Eur. J. Oper. Res. 202(1), 164–174 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Wan, X.T., Pekny, J.F., Reklaitis, G.V.: Simulation-based optimization with surrogate models—application to supply chain management. Comput. Chem. Eng. 29(6), 1317–1328 (2005)CrossRefGoogle Scholar
  13. 13.
    Espinet, A., Shoemaker, C., Doughty, C.: Estimation of plume distribution for carbon sequestration using parameter estimation with limited monitoring data. Water Resour. Res. 49(7), 4442–4464 (2013)CrossRefGoogle Scholar
  14. 14.
    Hasan, M.M.F., Baliban, R.C., Elia, J.A., Floudas, C.A.: Modeling, simulation, and optimization of postcombustion CO\(_{2}\) capture for variable feed concentration and flow rate. 2. Pressure swing adsorption and vacuum swing adsorption processes. Ind. Eng. Chem. Res. 51(48), 15665–15682 (2013)CrossRefGoogle Scholar
  15. 15.
    Hasan, M.M.F., Boukouvala, F., First, E.L., Floudas, C.A.: Nationwide, regional, and statewide \(CO_{2}\) capture, utilization, and sequestration supply chain network optimization. Ind. Eng. Chem. Res. 53(18), 7489–7506 (2014)CrossRefGoogle Scholar
  16. 16.
    Li, S., Feng, L., Benner, P., Seidel-Morgenstern, A.: Using surrogate models for efficient optimization of simulated moving bed chromatography. Comput. Chem. Eng. 67, 121–132 (2014)CrossRefGoogle Scholar
  17. 17.
    Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling—A Practical Guide. Wiley, Chichester (2008)CrossRefGoogle Scholar
  18. 18.
    Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization, vol. 8. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Martelli, E., Amaldi, E.: PGS-COM: a hybrid method for constrained non-smooth black-box optimization problems: brief review, novel algorithm and comparative evaluation. Comput. Chem. Eng. 63, 108–139 (2014)CrossRefGoogle Scholar
  20. 20.
    Rios, L.M., Sahinidis, N.V.: Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Global Optim. 56(3), 1247–1293 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Boukouvala, F., Misener, R., Floudas, C.A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO. Eur. J. Oper. Res. 252, 701–727(2016) (in press). doi: 10.1016/j.ejor.2015.12.018
  22. 22.
    Bjorkman, M., Holmstrom, K.: Global optimization of costly nonconvex functions using radial basis functions. Optim. Eng. 1(4), 373–397 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Booker, A.J., Dennis, J.E., Frank, P.D., Serafini, D.B., Torczon, V., Trosset, M.W.: A rigorous framework for optimization of expensive functions by surrogates. Struct. Multidiscip. Optim. 17(1), 1–13 (1999)CrossRefGoogle Scholar
  24. 24.
    Boukouvala, F., Ierapetritou, M.G.: Derivative-free optimization for expensive constrained problems using a novel expected improvement objective function. AIChE J. 60(7), 2462–2474 (2014)CrossRefGoogle Scholar
  25. 25.
    Conn, A.R., Le Digabel, S.: Use of quadratic models with mesh-adaptive direct search for constrained black box optimization. Optim. Methods Softw. 28(1), 139–158 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1), 50–79 (2009)CrossRefGoogle Scholar
  27. 27.
    Jakobsson, S., Patriksson, M., Rudholm, J., Wojciechowski, A.: A method for simulation based optimization using radial basis functions. Optim. Eng. 11(4), 501–532 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jones, D.R.: A taxonomy of global optimization methods based on response surfaces. J. Global Optim. 21(4), 345–383 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Regis, R.G.: Constrained optimization by radial basis function interpolation for high-dimensional expensive black-box problems with infeasible initial points. Eng. Optim. 46(2), 218–243 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Regis, R.G., Shoemaker, C.A.: Constrained global optimization of expensive black box functions using radial basis functions. J. Global Optim. 31(1), 153–171 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yao, W., Chen, X.Q., Huang, Y.Y., van Tooren, M.: A surrogate-based optimization method with RBF neural network enhanced by linear interpolation and hybrid infill strategy. Optim. Methods Softw. 29(2), 406–429 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Muller, J., Shoemaker, C.A.: Influence ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization methods. J. Glob. Optim. 60(2), 123–144 (2014)CrossRefzbMATHGoogle Scholar
  34. 34.
    Viana, F.A.C., Haftka, R.T., Watson, L.T.: Efficient global optimization algorithm assisted by multiple surrogate techniques. J. Glob. Optim. 56(2), 669–689 (2013)CrossRefzbMATHGoogle Scholar
  35. 35.
    Floudas, C.A.: Deterministic Global Optimization, vol. 37. Springer Science & Business Media, New York (1999)Google Scholar
  36. 36.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Global Optim. 45(1), 3–38 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Boukouvala, F., Hasan, M.M.F., Christodoulos, A.F.: Global optimization of constrained grey-box problems: new method and application to constrained PDEs for pressure swing adsorption. J. Global Optim. (2015). doi: 10.1007/s10898-015-0376-2
  38. 38.
    Misener, R., Floudas, C.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Global Optim. 59(2–3), 503–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Global Optim. 7(2), 143–182 (1995)Google Scholar
  40. 40.
    Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Global Optim. 57(1), 3–50 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Li, Z., Floudas, C.A.: Optimal scenario reduction framework based on distance of uncertainty distribution and output performance: I. Single reduction via mixed integer linear optimization. Comput. Chem. Eng. 70, 50–65 (2014)CrossRefGoogle Scholar
  42. 42.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press (1981)Google Scholar
  43. 43.
    Holmstrom, K., Quttineh, N.-H., Edvall, M.M.: An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization. Optim. Eng. 9(4), 311–339 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kleijnen, J.P.C.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192(3), 707–716 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Adjengue, L., Audet, C., Ben Yahia, I.: A variance-based method to rank input variables of the mesh adaptive direct search algorithm. Optim. Lett. 8(5), 1599–1610 (2014). doi: 10.1007/s11590-013-0688-4 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Fan, J., Lv, J.: Sure independence screening for ultrahigh dimensional feature space. J. R. Stat. Soc. Ser. B (Statistical Methodology) 70(5), 849–911 (2008). doi: 10.1111/j.1467-9868.2008.00674.x MathSciNetCrossRefGoogle Scholar
  47. 47.
    Fan, J., Song, R.: Sure independence screening in generalized linear models with NP-dimensionality. Ann. Stat. 38(6) 3567–3604 (2010). doi: 10.1214/10-AOS798
  48. 48.
    Fan, J., Feng, Y., Saldana, D.F., Samworth, R., Wu, Y.: Sure independence screening version 0.7-5 (2015).
  49. 49.
    Misener, R., Floudas, C.A.: Global optimization of mixed-integer models with quadratic and signomial functions: a review. Appl. Comput. Math. 11, 317–336 (2012)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Berlin (1999)CrossRefzbMATHGoogle Scholar
  51. 51.
    Abramson, M.A., Audet, C., Couture, G., Dennis, J.E., S, Tribes, C.: The NOMAD Project (2009).
  52. 52.
    Le Digabel, S.: Algorithm 909: NOMAD: nonlinear optimization with the MADS algorithm. ACM Trans. Math. Softw. (TOMS) 37(4), 1–15 (2011)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Runarsson, T.P., Xin, Y.: Search biases in constrained evolutionary optimization. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 35(2), 233–243 (2005). doi: 10.1109/TSMCC.2004.841906 CrossRefGoogle Scholar
  54. 54.
    Johnson, S.G.: The NLopt nonlinear-optimization package (2015).
  55. 55.
    Powell, M.J.D.: A Direct Search Optimization Method that Models the Objective and Constraint Functions by Linear Interpolation. In: Gomez, S., Hennart, J.-P. (eds.) Advances in Optimization and Numerical Analysis. Mathematics and Its Applications, vol. 275, pp. 51–67. Springer, Netherlands (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Artie McFerrin Department of Chemical EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Texas A&M Energy InstituteTexas A&M UniversityCollege StationUSA

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