Skip to main content
Log in

Global optimization of non-convex piecewise linear regression splines

Journal of Global Optimization Aims and scope Submit manuscript

Cite this article


Multivariate adaptive regression spline (MARS) is a statistical modeling method used to represent a complex system. More recently, a version of MARS was modified to be piecewise linear. This paper presents a mixed integer linear program, called MARSOPT, that optimizes a non-convex piecewise linear MARS model subject to constraints that include both linear regression models and piecewise linear MARS models. MARSOPT is customized for an automotive crash safety system design problem for a major US automaker and solved using branch and bound. The solutions from MARSOPT are compared with those from customized genetic algorithms.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1


  1. Aspenberg, D., Jergues, J., Nilsson, L.: Robust optimization of front members in a full frontal car impact. Eng. Optim. 45, 245–264 (2013)

    Article  Google Scholar 

  2. Chen, S.Y.: An approach for impact structure optimization using the robust genetic algorithm. Finite Elem. Anal. Des. 37, 431–446 (2001)

    Article  MATH  Google Scholar 

  3. Crino, S., Brown, D.E.: Global optimization with multivariate adaptive regression splines. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 37(2), 333–340 (2007)

    Article  Google Scholar 

  4. Emmerich, M., Giotis, A., Ozdemir, M., Back, T., Giannakoglou, K.: Metamodel assisted evolution strategies. In: Parallel problem solving from nature V (PPSN VII), Springer, Berlin, pp. 362–370 (2002)

  5. Friedman, J.H.: Multivariate adaptive regression splines (with discussion). Ann. Stat. 19, 1–141 (1991)

    Article  MATH  Google Scholar 

  6. Glover, F.: Heuristics for integer programming using surrogate constraints. Decis. Sci. 8, 156–166 (1977)

    Article  Google Scholar 

  7. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley, Reading (1989)

    MATH  Google Scholar 

  8. Grefenstette, J.J.: Optimization of control parameters for genetic algorithms. IEEE Trans. Syst. Man Cybern. 16, 122–128 (1986)

    Article  Google Scholar 

  9. Gu, L.: A comparison of polynomial based regression models in vehicle safety analysis. In: ASME design engineering technical conferences—design automation (2001)

  10. Gu, L., Yang, R., Tho, C., Makowskit, M., Faruquet, O., Li, Y.: Optimization and robustness for crashworthiness of side impact. Int. J. Veh. Des. 26(4), 348–360 (2001)

    Article  Google Scholar 

  11. Gutmann, H.M.: A radial basis function method for global optimization. J. Glob. Optim. 19(3), 201–227 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamza, K., Saitou, K.: A co-evolutionary approach for design optimization via ensembles of surrogates with application to vehicle crashworthiness. J. Mech. Des. 134(1), 011001-1–011001-10 (2012)

    Article  Google Scholar 

  13. Holland, J.H.: Adaptation in natural and artificial systems. The University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  14. Horst, R., Pardalos, P.M., Thoai, N.: Introduction to Global Optimization. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  15. Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Keha, A.B., de Farias, I.R., Nemhauser, G.L.: Models for representing piecewise linear cost functions. Oper. Res. Lett. 32, 44–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Keha, A.B., de Farias, I.R., Nemhauser, G.L.: A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54, 847–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kirkpatrick, S., Gelatt, C., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kutner, M.H.: Applied Linear Statistical Model. McGraw-HilL/Irwin, NY (1974)

    Google Scholar 

  20. Liao, X., Qing, L., Xujing, Y., Weigang, Z., Wei, L.: Multiobjective optimization for crash safety design of vehicles using stepwise regression model. Struct. Multidiscip. Optim. 35(6), 561–569 (2008)

    Article  Google Scholar 

  21. Martinez, D.: Variants of Multivariate Adaptive Regression Splines (mars): Convex vs. Non-convex, Piecewise-Linear vs. Smooth and Sequential Algorithms. Ph.D. thesis, The University of Texas at Arlington (2013)

  22. Martinez, D., Shih, T.D., Chen, V.C.P., Kim, S.B.: A convex version of multivariate adaptive regression splines. Comput. Stat. Data Anal. 81, 89–106 (2015)

    Article  MathSciNet  Google Scholar 

  23. Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, Third, Revised and Extended Edition (1996)

  24. Mohamed, A.W., Sabry, H.Z., Khorshid, M.: An alternative differential evolution algorithm for global optimization. J. Adv. Res. 3, 149–165 (2012)

    Article  Google Scholar 

  25. National Highway Traffic Safety Administration: 5-star safety ratings frequently asked questions (2016).

  26. Peremezhney, N., Hines, E., Lapkin, A., Connaughton, C.: Combining gaussian processes, mutual information and a genetic algorithm for multi-target optimization of expensive-to-evaluate functions. Eng. Optim. 46(11), 1593–1607 (2014)

    Article  MathSciNet  Google Scholar 

  27. Pilla, V.L., Rosenberger, J.M., Chen, V.C.P., Smith, B.C.: A statistical computer experiments approach to airline fleet assignment. IIE Trans. COSMOS Technical Report 05-03, vol. 40(5), pp. 524–537. The University of Texas at Arlington (2008)

  28. Regis, R.G., Shoemaker, C.A.: A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J. Comput. 19(4), 497–509 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rich, E., Knight, K.: Artificial Intelligence. McGraw-Hill, NY (1991)

    Google Scholar 

  30. Romeijn, H.E., Pardalos, P.M.: Handbook of Global Optimization, vol. 2. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  31. Sherali, H.D., Ganesan, V.: A pseudo-global optimization approach with application to the design of containerships. J. Glob. Optim. 26, 335–360 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Glob. Optim. 2, 101–112 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sherali, H.D., Wang, H.: Global optimization of non-convex factorable programming problems. Math. Program. Ser. A 89, 459–478 (2001)

    Article  MATH  Google Scholar 

  34. Shih, T.D.: Convex versions of multivariate adaptive regression splines and implementation for complex optimization problems. Ph.D. thesis, Department of Industrial and System Engineering, UT Arlington (2006)

  35. Siddappa, S., Günther, D., Rosenberger, J., Chen, V.C.P.: Refined experimental design and regression splines method for network revenue management. J. Pricing Revenue Manag. 6(3), 188–199 (2007)

    Article  Google Scholar 

  36. Song, X., Li, G., Gao, W., Li, Q.: Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models. Struct. Multidiscip. Optim. 47(2), 221–231 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vielma, J., Keha, A., Nemhauser, G.: Nonconvex, lower semi-continuous piecewise linear optimization. Discrete Optim. 5, 467–488 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, G., Shan, S.: Review of metamodeling techniques in support of engineering design optimization. ASME Trans. J. Mech. Des. 129(4), 370–380 (2007)

    Article  Google Scholar 

  39. Wang, H., Li, G., Li, E.: Time-based metamodeling technique for vehicle crashworthiness optimization. Comput. Methods Appl. Mech. Eng. 199, 37–40 (2010)

    MATH  Google Scholar 

  40. Willmes, L., Back, T., Jin, Y., Sendhoff, B.: Comparing neural networks and kriging for fitness approximation in evolutionary optimization. In: IEEE congress on evolutionary computation, pp 663–670 (2003)

  41. Yang, R.J., Gu, L., Liaw, L., Gearhart, C., Tho, C.H., Liu, X., Wang, B.P.: Approximations for safety optimization of large systems. In: ASME 2000 design engineering technical conferences—design automation conference (2000)

  42. Yang, R.J., Wang, N., Tho, C.H., Bobineau, J.P., Wang, B.P.: Metamodeling development for vehicle frontal impact simulation. J. Mech. Des. 127(5), 1014–1020 (2005)

    Article  Google Scholar 

  43. Yin, H., Wen, G., Fang, H., Qing, Q., Kong, X., Xiao, J., Liu, Z.: Multiobjective crashworthiness optimization design of functionally graded foam-filled tapered tube based on dynamic ensemble metamodel. Mater. Des. 55, 747–757 (2014)

    Article  Google Scholar 

Download references


This research was partially supported by National Science Foundation Award CMMI–1434401.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hadis Anahideh.


Appendix 1

Table 9 displays the scaled and unscaled solutions found with MARSOPT using both the SLR model and the PL-MARS model for the objective function, while Table 10 reports the objective values and the output variables (left-hand sides of the constraints) for both solutions.

Table 9 Scaled and unscaled solutions obtained from MARSOPT using SLR model and PL-MARS model
Table 10 Objective value and output values for the constraints

Appendix 2

The unscaled solutions for Solutions 5, 14, and 15 found using MARSOPT and Solution 4 from Data Set 1 are displayed in Table 11.

Table 11 Unscaled solutions for selected points

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martinez, N., Anahideh, H., Rosenberger, J.M. et al. Global optimization of non-convex piecewise linear regression splines. J Glob Optim 68, 563–586 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: