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An approach to solve local and global optimization problems based on exact objective filled penalty functions
doi: 10.3934/jimo.2022016
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## A global optimization method for multiple response optimization problems

 College of Management and Economics, Tianjin University, Tianjin 300072, China

* Corresponding author: He Huang

Received  June 2021 Revised  December 2021 Early access February 2022

The multiple response optimization problem has been studied extensively. However, most existing methods only find locally optimal solutions to the concerned optimization problem. Several methods were proposed which tried to find a globally optimal solution of the problem, but there is no theoretical guarantee to obtain a globally optimal solution. In this paper, we investigate a global optimization method for the problem of a chemical process studied by Myers et al. which involves two input variables and three responses of interest. Based on the fitted polynomial functions of three responses, this multiple response problem is reformulated as a polynomial optimization problem where the primary response is objective while the other responses are put into constraints. We obtain a globally optimal solution to the concerned polynomial optimization problem when requirements of non-primary responses and experimental region are given. The satisfactory optimal designs can be obtained by adjusting non-primary responses appropriately. The method we proposed can be implemented easily, and it obtains a globally optimal solution to the multiple response optimization problem we considered.

Citation: He Huang, Zhen He. A global optimization method for multiple response optimization problems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022016
##### References:
 [1] Ç. S. Aksezer, On the sensitivity of desirability functions for multiresponse optimization, Journal of Industrial and Management Optimization, 4 (2008), 685-696.  doi: 10.3934/jimo.2008.4.685. [2] W. E. Biles, A response surface methods for experimental optimization of multi-response processes, Industrial and Engineering Chemistry Process Design and Deployment, 14 (1975), 152-158.  doi: 10.1021/i260054a010. [3] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Process optimization for multiple responses utilizing the Pareto front approach, Quality Engineering, 26 (2014), 253-268.  doi: 10.1080/08982112.2013.852681. [4] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Incorporating response variability and estimation uncertainty into Pareto front optimization, Computers & Industrial Engineering, 76 (2014), 253-267.  doi: 10.1016/j.cie.2014.07.028. [5] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Impact of response variability on Pareto front optimization, Statistical Analysis & Data Mining, 8 (2015), 314-328.  doi: 10.1002/sam.11279. [6] R. Curto and L. Fialkow, Truncated $K$-moment problems in several variables, Journal of Operator Theory, 54 (2005), 189–226, https://www.jstor.org/stable/24715679. [7] E. del Castillo, Multiresponse process optimization via constrained confidence regions, Journal of Quality Technology, 28 (1996), 61-70.  doi: 10.1080/00224065.1996.11979637. [8] G. Derringer and R. Suich, Simultaneous optimization of several response variables, Journal of Quality Thchnology, 12 (1980), 214-219.  doi: 10.1080/00224065.1980.11980968. [9] Z. He, P. F. Zhu and H. S. Park, A robust desirability function method for multi-response surface optimization considering model uncertainty, European Journal of Operational Research, 221 (2012), 241-247.  doi: 10.1016/j.ejor.2012.03.009. [10] E. C. Jr Harrington, The desriability function, Industrial Quality Control, 21 (1965), 494-498. [11] A. I. Khuri, 12 Multiresponse surface methodology, in Handbook of Statistics: Design and Analysis of Experiments (eds. A. Ghosh and C. R. Rao), Elsevier, Amsterdam, 13 (1996), 377–406. doi: 10.1016/S0169-7161(96)13014-5. [12] A. I. Khuri and J. A. Cornell, Response Surfaces: Designs and Analyses (2th edn), Dekker, New York, 1996. [13] J. B. Lasserre, Global optimization with polynomial and the problem of moments, SIAM Journal on Optimization, 11 (2001), 796-817.  doi: 10.1137/S1052623400366802. [14] J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, World Scientific, Singapore, 2010. [15] J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. [16] J. Löfberg, Pre- and post-processing sums-of-squares programs in practice, IEEE Transactions on Automatic Control, 54 (2009), 1007-1011.  doi: 10.1109/TAC.2009.2017144. [17] N. Logothetis and H. P. Wynn, Quality Through Design, Oxford Science Publications, Clarendon Press, Oxford, 1989. [18] D. C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, New York-London-Sydney, 1976. [19] R. H. Myers and W. H. Carter, Response surface techniques for dual response systems, Technometrics, 15 (1973), 301-317.  doi: 10.1080/00401706.1973.10489044. [20] R. H. Myers and D. C. Montgomery, Response Surface Methodology (2nd edn), John Wiley & Sons, New York, 2002. [21] R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, Hoboken, NJ: Wiley, 2009. [22] J. Nie, Certifying convergence of Lasserre's hierachy via flat truncation, Mathematical Programming, 142 (2013), 485-510.  doi: 10.1007/s10107-012-0589-9. [23] L. Ouyang, Y. Ma and J. H. Byun, An integrative loss function approach to multi-response optimization, Quality and Reliability Engineering International, 31 (2015), 193-204.  doi: 10.1002/qre.1571. [24] L. Ouyang, Y. Ma, J. H. Byun, J. Wang and Y. Tu, A prediction region-based approach to model uncertainty for multi-response optimization, Quality and Reliability Engineering International, 32 (2016), 783-794.  doi: 10.1002/qre.1790. [25] K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 – a Matlab software package for semidefinite programming, Version 1.3, Interior point methods, Optimization Methods and Software, 11 (1999), 545-581.  doi: 10.1080/10556789908805762.

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##### References:
 [1] Ç. S. Aksezer, On the sensitivity of desirability functions for multiresponse optimization, Journal of Industrial and Management Optimization, 4 (2008), 685-696.  doi: 10.3934/jimo.2008.4.685. [2] W. E. Biles, A response surface methods for experimental optimization of multi-response processes, Industrial and Engineering Chemistry Process Design and Deployment, 14 (1975), 152-158.  doi: 10.1021/i260054a010. [3] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Process optimization for multiple responses utilizing the Pareto front approach, Quality Engineering, 26 (2014), 253-268.  doi: 10.1080/08982112.2013.852681. [4] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Incorporating response variability and estimation uncertainty into Pareto front optimization, Computers & Industrial Engineering, 76 (2014), 253-267.  doi: 10.1016/j.cie.2014.07.028. [5] J. L. Chapman, L. Lu and C. M. Anderson-Cook, Impact of response variability on Pareto front optimization, Statistical Analysis & Data Mining, 8 (2015), 314-328.  doi: 10.1002/sam.11279. [6] R. Curto and L. Fialkow, Truncated $K$-moment problems in several variables, Journal of Operator Theory, 54 (2005), 189–226, https://www.jstor.org/stable/24715679. [7] E. del Castillo, Multiresponse process optimization via constrained confidence regions, Journal of Quality Technology, 28 (1996), 61-70.  doi: 10.1080/00224065.1996.11979637. [8] G. Derringer and R. Suich, Simultaneous optimization of several response variables, Journal of Quality Thchnology, 12 (1980), 214-219.  doi: 10.1080/00224065.1980.11980968. [9] Z. He, P. F. Zhu and H. S. Park, A robust desirability function method for multi-response surface optimization considering model uncertainty, European Journal of Operational Research, 221 (2012), 241-247.  doi: 10.1016/j.ejor.2012.03.009. [10] E. C. Jr Harrington, The desriability function, Industrial Quality Control, 21 (1965), 494-498. [11] A. I. Khuri, 12 Multiresponse surface methodology, in Handbook of Statistics: Design and Analysis of Experiments (eds. A. Ghosh and C. R. Rao), Elsevier, Amsterdam, 13 (1996), 377–406. doi: 10.1016/S0169-7161(96)13014-5. [12] A. I. Khuri and J. A. Cornell, Response Surfaces: Designs and Analyses (2th edn), Dekker, New York, 1996. [13] J. B. Lasserre, Global optimization with polynomial and the problem of moments, SIAM Journal on Optimization, 11 (2001), 796-817.  doi: 10.1137/S1052623400366802. [14] J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, World Scientific, Singapore, 2010. [15] J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. [16] J. Löfberg, Pre- and post-processing sums-of-squares programs in practice, IEEE Transactions on Automatic Control, 54 (2009), 1007-1011.  doi: 10.1109/TAC.2009.2017144. [17] N. Logothetis and H. P. Wynn, Quality Through Design, Oxford Science Publications, Clarendon Press, Oxford, 1989. [18] D. C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, New York-London-Sydney, 1976. [19] R. H. Myers and W. H. Carter, Response surface techniques for dual response systems, Technometrics, 15 (1973), 301-317.  doi: 10.1080/00401706.1973.10489044. [20] R. H. Myers and D. C. Montgomery, Response Surface Methodology (2nd edn), John Wiley & Sons, New York, 2002. [21] R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology, Hoboken, NJ: Wiley, 2009. [22] J. Nie, Certifying convergence of Lasserre's hierachy via flat truncation, Mathematical Programming, 142 (2013), 485-510.  doi: 10.1007/s10107-012-0589-9. [23] L. Ouyang, Y. Ma and J. H. Byun, An integrative loss function approach to multi-response optimization, Quality and Reliability Engineering International, 31 (2015), 193-204.  doi: 10.1002/qre.1571. [24] L. Ouyang, Y. Ma, J. H. Byun, J. Wang and Y. Tu, A prediction region-based approach to model uncertainty for multi-response optimization, Quality and Reliability Engineering International, 32 (2016), 783-794.  doi: 10.1002/qre.1790. [25] K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 – a Matlab software package for semidefinite programming, Version 1.3, Interior point methods, Optimization Methods and Software, 11 (1999), 545-581.  doi: 10.1080/10556789908805762.
Graph of the response yield
Graph of the response viscosity
Graph of the response molecular weight
Graph of the feasible region
The feasible region for different parts
Designed experiment and response values
 Order Natural variables Coded variables Responses $\xi_1$ $\xi_2$ $x_1$ $x_2$ $y_1$ $y_2$ $y_3$ 1 80 170 -1 -1 76.5 62 2940 2 90 170 1 -1 78.0 66 3680 3 80 180 -1 1 77.0 60 3470 4 90 180 1 1 79.5 59 3890 5 77.93 175 -1.414 0 75.6 71 3020 6 92.07 175 1.414 0 78.4 68 3360 7 85 167.93 0 -1.414 77.0 57 3150 8 85 182.07 0 1.414 78.5 58 3630 9 85 175 0 0 79.9 72 3480 10 85 175 0 0 80.3 69 3200 11 85 175 0 0 80.0 68 3410 12 85 175 0 0 79.7 70 3290 13 85 175 0 0 79.8 71 3500
 Order Natural variables Coded variables Responses $\xi_1$ $\xi_2$ $x_1$ $x_2$ $y_1$ $y_2$ $y_3$ 1 80 170 -1 -1 76.5 62 2940 2 90 170 1 -1 78.0 66 3680 3 80 180 -1 1 77.0 60 3470 4 90 180 1 1 79.5 59 3890 5 77.93 175 -1.414 0 75.6 71 3020 6 92.07 175 1.414 0 78.4 68 3360 7 85 167.93 0 -1.414 77.0 57 3150 8 85 182.07 0 1.414 78.5 58 3630 9 85 175 0 0 79.9 72 3480 10 85 175 0 0 80.3 69 3200 11 85 175 0 0 80.0 68 3410 12 85 175 0 0 79.7 70 3290 13 85 175 0 0 79.8 71 3500
The numerical results of problem (16)
 $Num$ $Va$ $Vb$ $Vc$ $Vd$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 -1.6 0.0 -1.6 0.0 0.0 -0.6224 79.229 68 3275.8 2 -1.6 0.0 0.0 1.6 -0.3710 0.5068 79.341 68 3400 3 0.0 1.6 -1.6 0.0 0.2789 -0.6390 79.325 68 3330.1 4 0.0 1.6 1.6 0.0 F F F F F 5 -1.6 1.6 -1.6 1.6 -0.3710 0.5068 79.341 68 3400
 $Num$ $Va$ $Vb$ $Vc$ $Vd$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 -1.6 0.0 -1.6 0.0 0.0 -0.6224 79.229 68 3275.8 2 -1.6 0.0 0.0 1.6 -0.3710 0.5068 79.341 68 3400 3 0.0 1.6 -1.6 0.0 0.2789 -0.6390 79.325 68 3330.1 4 0.0 1.6 1.6 0.0 F F F F F 5 -1.6 1.6 -1.6 1.6 -0.3710 0.5068 79.341 68 3400
The numerical results (Ⅰ) of problem (17)
 $Num$ $\alpha$ $\beta_1$ $\beta_2$ $\gamma$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 N N N N 0.3885 0.3085 80.213 68.7539 3520.6 2 N N N 3500 0.3389 0.2496 80.207 69.1067 3500 3 N N N 3400 0.0987 -0.0363 80.004 70.0076 3400 4 N N N 3300 -0.1416 -0.3222 79.511 69.5634 3300 5 N N N 3200 -0.3819 -0.6081 78.731 67.7740 3200 6 N N N 3100 -0.6220 -0.8942 77.662 64.6379 3100 7 N N 68 N 0.4130 0.4228 80.2 68 3545.9 8 N N 67 N 0.4372 0.5445 80.157 67 3572.5 9 N N 66 N 0.4566 0.64698 80.098 66 3594.6 10 N N 65 N 0.4733 0.7370 80.029 65 3614 11 N N 64 N 0.4881 0.8184 79.953 64 3631.5 12 N N 63 N 0.5016 0.8932 79.871 63 3647.5 13 N N 62 N 0.5140 0.9628 79.784 62 3662.4
 $Num$ $\alpha$ $\beta_1$ $\beta_2$ $\gamma$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 N N N N 0.3885 0.3085 80.213 68.7539 3520.6 2 N N N 3500 0.3389 0.2496 80.207 69.1067 3500 3 N N N 3400 0.0987 -0.0363 80.004 70.0076 3400 4 N N N 3300 -0.1416 -0.3222 79.511 69.5634 3300 5 N N N 3200 -0.3819 -0.6081 78.731 67.7740 3200 6 N N N 3100 -0.6220 -0.8942 77.662 64.6379 3100 7 N N 68 N 0.4130 0.4228 80.2 68 3545.9 8 N N 67 N 0.4372 0.5445 80.157 67 3572.5 9 N N 66 N 0.4566 0.64698 80.098 66 3594.6 10 N N 65 N 0.4733 0.7370 80.029 65 3614 11 N N 64 N 0.4881 0.8184 79.953 64 3631.5 12 N N 63 N 0.5016 0.8932 79.871 63 3647.5 13 N N 62 N 0.5140 0.9628 79.784 62 3662.4
The numerical results (Ⅱ) of problem (17)
 $Num$ $\alpha$ $\beta_1$ $\beta_2$ $\gamma$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 N N 68 3400 -0.3710 0.5068 79.341 68 3400 2 N N 68 3300 0.1260 -0.6315 79.296 68 3300 3 N N 68 3200 -0.3817 -0.6083 78.731 67.7726 3200 4 N N 67 3400 0.2537 -0.7616 79.079 67 3303.1 5 N N 67 3300 0.2377 -0.7607 79.079 67 3300 6 N N 67 3200 -0.2906 -0.7136 78.706 67 3200 7 N N 66 3400 0.2338 -0.8644 78.850 66 3280.8 8 N N 66 3200 -0.1910 -0.8287 78.621 66 3200 9 N N 65 3400 0.2168 -0.9548 78.631 65 3261.3 10 N N 65 3200 -0.1046 -0.9287 78.5 65 3200 11 N N 64.9 3400 0.2152 -0.9633 78.610 64.9 3259.5 12 N N 64.9 3200 -0.0965 -0.9381 78.486 64.9 3200 13 N N 64 3400 0.2018 -1.0364 78.419 64 3243.7 14 N N 64 3200 -0.0271 -1.0183 78.353 64 3200
 $Num$ $\alpha$ $\beta_1$ $\beta_2$ $\gamma$ $x_1^*$ $x_2^*$ $\hat{y}_1^*$ $\hat{y}_2^*$ $\hat{y}_3^*$ 1 N N 68 3400 -0.3710 0.5068 79.341 68 3400 2 N N 68 3300 0.1260 -0.6315 79.296 68 3300 3 N N 68 3200 -0.3817 -0.6083 78.731 67.7726 3200 4 N N 67 3400 0.2537 -0.7616 79.079 67 3303.1 5 N N 67 3300 0.2377 -0.7607 79.079 67 3300 6 N N 67 3200 -0.2906 -0.7136 78.706 67 3200 7 N N 66 3400 0.2338 -0.8644 78.850 66 3280.8 8 N N 66 3200 -0.1910 -0.8287 78.621 66 3200 9 N N 65 3400 0.2168 -0.9548 78.631 65 3261.3 10 N N 65 3200 -0.1046 -0.9287 78.5 65 3200 11 N N 64.9 3400 0.2152 -0.9633 78.610 64.9 3259.5 12 N N 64.9 3200 -0.0965 -0.9381 78.486 64.9 3200 13 N N 64 3400 0.2018 -1.0364 78.419 64 3243.7 14 N N 64 3200 -0.0271 -1.0183 78.353 64 3200
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