Nondifferentiable Desirability Functions: Derivative Free Optimization with MATLAB/NOMAD

Abstract

Nondifferentiable Desirability functions are one of the most preferred multi-response optimiza- tion methods in nonlinear robust parameter design. Their nondifferentiability makes the optimization prob- lem hard to solve and researchers and scientists look for new softwares and new desirability function structures to overcome this problem. In this study, we suggest a new implementation of derivative free mash adaptive direct search algorithm (MADS) with MATLAB/NOMAD to nondifferentiable desirability functions. For doing this, we need to model the optimization problem of desirability functions as a mixed- integer nonlinear optimization program (MINLP) by introducing a new binary variable to the model. This integer shows the side of the two-sided desirability function which is active. Hence, the model of our problem becomes nondifferentiable nonconvex MINLP. We show our implementation on three well-known optimization problem from the multi-response optimization literature. We finally conclude with an outlook and future research projects.

Keywords

• derivative free optimization
• multi-response optimization
• matlab

References

1. B. Akteke-O¨ ztu¨rk, G. Ko¨ksal, G. W. Weber, Nonconvex Optimization Desirability Functions, Quality Engineering 30(2) (2017) 293–310.
2. B. Akteke-O¨ ztu¨rk, G. W. Weber, G. Ko¨ksal, Optimization of Generalized Desirability Functions under Model Uncertainty, Optimization, 66(12) (2017) 2157–2169.
3. Akteke-O¨ ztu¨rk, B., Weber, G.-W., Ko¨ksal, G. (2015). Desirability Functions in Multiresponse Optimization. In: Plakhov A., Tchemisova T., Freitas A. (Eds.) Optimization in the Natural Sciences. EmC-ONS 2014. Communications in Computer and Information Science, 499: 129-146. Springer.
4. B. Akteke-O¨ ztu¨rk, G. W. Weber, G. Ko¨ksal, Generalized Desirability Functions: A Structural and Topological Analysis of Desirability Functions, Optimization 69(1) (2019) 115–130.
5. B. Akteke-O¨ ztu¨rk, Mixed-Integer Linear Programming Approaches for Nonlinear Robust Parameter Design Problems: A Study, submitted.
6. BARON (2010). www.gams.com/solver, v. 8.1.5.
7. P. Belotti, C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, A. Mahajan, Mixed-integer nonlinear optimization, Acta Numerica 22 (2013) 1–131.
8. G.E.P. Box, D.W. Behnken, Some new three-level designs for the study of quantitative variables, Technometrics 2 (1960): 455-475.

9. E. Del Castillo, D. C. Montgomery, D. R. McCarville, Modified desirability functions for multiple response optimization, Journal of Quality Technology, 28(3) (1996) 337–345.
10. A. R. Conn, K. Scheinberg, L. N. Vicente, Introduction to derivative free optimization, MOS-SIAM Series on Optimization (2009).
11. CONOPT (2010). www.gams.com/solver, v. 3.14S.
12. CPLEX (2012). www.gams.com/solver, v. 12.4.
13. G. Derringer, R. Suich, Simultaneous optimization of several response variables, Journal of Quality Technology, 12 (1980) 214–219.
14. G. Derringer, A balancing act, Optimizing a products properties, Quality Progress, 27 (1994) 51–57.
15. GAMS (2012). www.gams.com, v. 23.8.2.
16. E.C.Jr. Harrington, The desirability function, Industrial Quality Control, 21 (1965) 494–498.
17. Z. He, P.F. Zhu, S.H. Park, A robust desirability function method for multi-response surface optimization considering model uncertainty. European Journal of Operational Research 221 (1) (2012) 241–247.
18. I.J. Jeong, K.J. Kim, An interactive desirability function method to multiresponse optimization. European Journal of Operational Research, 195 (2) (2008) 412-426.
19. I.J. Jeong, K.J. Kim, D-STEM: a modified step method with desirability function concept, Computers and Operations Research 32 (2005) 3175–3190.
20. A.I. Khuri, Multiresponse surface methodology, In: Ghosh, A., and Rao, C.R. (Eds.). Handbook of Statistics: Design and Analysis of Experiments, 13 (1996) 377–406.
21. K. Kim, D. Lin, Simultaneous optimization of multiple responses by maximizing exponential desirability functions. Applied Statistics, 49(C) (2000) 311-3-25.
22. O. K¨oksoy, Dual response optimization: The desirability approach, International Journal of Industrial Engineering, 12(4) (2005) 335–342.
23. D.H. Lee, K.J. Kim, M. K¨oksalan, A posterior preference articulation approach to multiresponse surface optimization, European Journal of Operational Research, 210 (2011) 301–309.
24. D.H. Lee, K.J. Kim, M. K¨oksalan, An interactive method to multiresponse surface optimization based on pairwise comparisons, IIE Transactions, 44(1) (2012) 13–26.
25. R.T. Marler, J.S. Arora, Survey of multi-objectivee optimization methods for engineering, Structural and Multidisciplinary Optimization, 26(6) (2004) 369–395.
26. the Mathworks Inc. (2025). www.mathworks.com. v. R2025a.
27. K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers (1999).
28. G. Nemhauser, L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley and Sons, New York (1988).
29. S. L. Digabel, Algorithm 909: NOMAD: Nonlinear optimization with the MADS algorithm, ACM Transactions on Mathematical Software, 37(4): 44:1-44:15.
30. K.S. Park, K.J. Kim, Optimizing multi-response surface problems: How to use multi-objective optimization techniques, IIE Transactions 37(6) (2005) 523–532.
31. R. P¨orn, K.M. Bj¨ork, T. Westerlund, Global solution of optimization problems with signomial parts, Discrete Optimization, 5 (2008) 108–120.
32. G. Taguchi, Introduction to Quality Engineering: Designing Quality into Products and Processes, Kraus, White Plains, NY (1986).
33. G. Taguchi, System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Cost, UNIPUB/Kraus International, White Plains, NY (1987).
34. H.P. Williams, Model Building in Mathematical Programming, fifth edition, Wiley (2013).
35. O. P. Yadav, G. Thambidorai, B. Nepal, L. Monplaisir, A robust framework for multi-response surface optimization methodology, Quality Reliability Engineering International. 30 (2014) 301–311.