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An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming

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Abstract

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.

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Acknowledgements

This work is in part supported by the AFRL Mathematical Modeling and Optimization Institute, and National Science Foundation through Grant CMMI-1355939. The authors would also like to thank the reviewers and the Editor for their helpful suggestions and comments.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL, 32816, USA

    Yiduo Zhan & Qipeng P. Zheng

  2. UNSW Business School, UNSW, Sydney, NSW, 2052, Australia

    Chung-Li Tseng

  3. Air Force Research Laboratory, 101 West Eglin Boulevard, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

Authors
  1. Yiduo Zhan
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  2. Qipeng P. Zheng
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  3. Chung-Li Tseng
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Corresponding author

Correspondence to Qipeng P. Zheng.

Appendix: Details of the test problems and results

Appendix: Details of the test problems and results

For each test problem, we report the problem statement and the obtained optimal solutions for solving the upper bound problem \((\bar{P}_\epsilon )\) and the lower bound problem \((\underline{P}_\epsilon )\) in Tables 4, 5, 6, 7 and 8. We define the optimality gap as \((\bar{f}_0-\underline{f}_0)/\underline{f}_0\), where \(\bar{f}_0\) and \(\underline{f}_0\) are the obtained upper bound and lower bound for \(f_0\), respectively, to reflect the quality of the obtained solutions. Our obtained optimal solutions for solving the upper bound problems \((\bar{P}_\epsilon )\) are always feasible, which are used to compare with benchmarks. The violated constraints are highlighted in bold face in each table to help observe the feasibility of the solutions provided by different methods.

Example 1

Heat exchange design (Problem 5 in Dembo [7])

$$\begin{aligned} f_0(\mathbf{z})= & {} z_1+z_2+z_3 \\ f_1(\mathbf{z})= & {} 833.33252z_1^{-1}z_4z_6^{-1}+100z_6^{-1}-83{,}333.333z_1^{-1}z_6^{-1} \\ f_2(\mathbf{z})= & {} 1250z_2^{-1}z_5z_7^{-1}+z_4z_7^{-1} -1250z_2^{-1}z_4z_7^{-1} \\ f_3(\mathbf{z})= & {} 1{,}250{,}000z_3^{-1}z_8^{-1}+ z_5z_8^{-1} -2500z_3^{-1}z_5z_8^{-1} \\ f_4(\mathbf{z})= & {} 0.0025z_4+0.0025z_6 \\ f_5(\mathbf{z})= & {} 0.0025z_5+0.0025z_7-0.0025z_4 \\ f_6(\mathbf{z})= & {} 0.01z_8 -0.01z_5 \\ \mathbf{z}^L= & {} \{10^2,10^3,10^3,10,10,10,10,10\} \\ \mathbf{z}^U= & {} \{10^4, 10^4, 10^4, 10^3, 10^3, 10^3, 10^3, 10^3\} \end{aligned}$$
Table 4 Summary of test results for Example 1
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Table 5 Summary of test results for Example 2
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Table 6 Summary of test results for Example 3
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Table 7 Summary of test results for Example 4
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Table 8 Summary of test results for Example 5
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Example 2

A 3-stage membrane separation process (Problem 6 in Dembo [7])

$$\begin{aligned} f_0(\mathbf{z})= & {} z_{11}+z_{12}+z_{13} \\ f_1(\mathbf{z})= & {} 1.262626z_8z_{11}^{-1}-1.231059z_1z_8z_{11}^{-1} \\ f_2(\mathbf{z})= & {} 1.262626z_9z_{12}^{-1}-1.231059z_2z_9z_{12}^{-1} \\ f_3(\mathbf{z})= & {} 1.262626z_{10}z_{13}^{-1}-1.231059z_3z_{10}z_{13}^{-1} \\ f_4(\mathbf{z})= & {} 0.03475z_2z_5^{-1}+0.975z_2-0.00975z_2^2z_5^{-1} \\ f_5(\mathbf{z})= & {} 0.03475z_3z_6^{-1}+0.975z_3-0.00975z_3^2z_6^{-1} \\ f_6(\mathbf{z})= & {} z_1z_5^{-1}z_7^{-1}z_8+z_4z_5^{-1}-z_4z_5^{-1}z_7^{-1}z_8 \\ f_7(\mathbf{z})= & {} 0.002z_2z_9 +0.002z_5z_8+z_6+z_5-0.002z_1z_8-0.002z_6z_9 \\ f_8(\mathbf{z})= & {} z_2^{-1}z_3z_9^{-1}z_{10}+z_2^{-1}z_6+500z_9^{-1}-z_9^{-1}z_{10}-500z_2^{-1}z_6z_9^{-1} \\ f_9(\mathbf{z})= & {} 0.9z_2^{-1} + 0.002z_{10}-0.002z_2^{-1}z_3z_{10} \\ f_{10}(\mathbf{z})= & {} z_2z_3^{-1} \\ f_{11}(\mathbf{z})= & {} z_1z_2^{-1} \\ f_{12}(\mathbf{z})= & {} 0.002z_7 -0.002z_8 \\ f_{13}(\mathbf{z})= & {} 0.03475z_1z_4^{-1}+0.975z_1-0.00975z_1^2z_4^{-1} \\ \mathbf{z}^L= & {} \{0.1, 0.1, 0.9, 10^{-3}, 0.1, 0.1, 0.1, 0.1, 500, 0.1,1,10^{-3},10^{-3}\} \\ \mathbf{z}^U= & {} \{1, 1, 1, 0.1, 0.9, 0.9, 10^3, 10^3, 10^3, 500, 150, 150, 150\} \end{aligned}$$

Example 3

A 5-stage membrane separation process (Problem 7 in Dembo [7])

$$\begin{aligned} f_0(\mathbf{z})= & {} 1.262626(z_{12}+z_{13}+z_{14}+z_{15}+z_{16})\\&-1.23106(z_{1}z_{12}+z_{2}z_{13}+z_{3}z_{14}+z_{4}z_{15}+z_{5}z_{16}) \\ f_1(\mathbf{z})= & {} 0.03475z_1z_{6}^{-1}+0.975z_1 -0.00975z_1^{2}z_{6}^{-1} \\ f_2(\mathbf{z})= & {} 0.03475z_2z_{7}^{-1}+0.975z_2-0.00975z_2^{2}z_{7}^{-1} \\ f_3(\mathbf{z})= & {} 0.03475z_3z_{8}^{-1}+0.975z_3-0.00975z_3^{2}z_{8}^{-1} \\ f_4(\mathbf{z})= & {} 0.03475z_4z_{9}^{-1}+0.975z_4-0.00975z_4^{2}z_{9}^{-1} \\ f_5(\mathbf{z})= & {} 0.03475z_5z_{10}^{-1}+0.975z_5-0.00975z_5^{2}z_{10}^{-1} \\ f_6(\mathbf{z})= & {} z_6z_7^{-1}+z_1z_7^{-1}z_{11}^{-1}z_{12}-z_6z_7^{-1}z_{11}^{-1}z_{12} \\ f_7(\mathbf{z})= & {} z_7z_8^{-1}+0.002z_7z_8^{-1}z_{12}+0.002z_2z_8^{-1}z_{13} -0.002z_{13}-0.002z_1z_8^{-1}z_{12}\\ f_8(\mathbf{z})= & {} z_8+0.002z_8z_{13}+0.002z_3z_{14}+z_9-0.002z_2z_{13}-0.002z_9z_{14}\\ f_9(\mathbf{z})= & {} z_3^{-1}z_9+z_3^{-1}z_4z_{14}^{-1}z_{15}+500z_3^{-1}z_{10}z_{14}^{-1} -500z_3^{-1}z_9z_{14}^{-1} \\&-z_3^{-1}z_8z_{14}^{-1}z_{15}\\ f_{10}(\mathbf{z})= & {} z_4^{-1}z_5z_{15}^{-1}z_{16}+z_4^{-1}z_{10} +500z_{15}^{-1}-z_{15}^{-1}z_{16}-500z_4^{-1}z_{10}z_{15}^{-1} \\ f_{11}(\mathbf{z})= & {} 0.9z_4^{-1}+0.002z_{16}-0.002z_4^{-1}z_5z_{16} \\ f_{12}(\mathbf{z})= & {} 0.002z_{11}-0.002z_{12} \\ f_{13}(\mathbf{z})= & {} z_{11}^{-1}z_{12}\\ f_{14}(\mathbf{z})= & {} z_{4}z_{5}^{-1}\\ f_{15}(\mathbf{z})= & {} z_{3}z_{4}^{-1}\\ f_{16}(\mathbf{z})= & {} z_{2}z_{3}^{-1}\\ f_{17}(\mathbf{z})= & {} z_{1}z_{2}^{-1}\\ f_{18}(\mathbf{z})= & {} z_{9}z_{10}^{-1}\\ f_{19}(\mathbf{z})= & {} z_{8}z_{9}^{-1} \\ \mathbf{z}^L= & {} \{0.1, 0.1, 0.1, 0.1,0.9, 10^{-4}, 0.1, 0.1, 0.1, 0.1, 1, 10^{-6},1,500,500,10^{-5}\} \\ \mathbf{z}^U= & {} \{0.9,0.9,0.9,0.9,1, 0.1, 0.9, 0.9,0.9,0.9, 10^3, 500, 500, 10^3, 10^3, 500\} \end{aligned}$$

Example 4

(Problem 5 in [23])

$$\begin{aligned} f_0(\mathbf{z})= & {} 2z_1^{0.9}z_2^{-1.5}z_3^{-3}+4.7z_6^{-1.8}z_7^{-0.5}z_8 +5z_4^{-0.3}z_5^{2.6}\\ f_1(\mathbf{z})= & {} 7.2z_1^{-3.8}z_2^{2.2}z_3^{4.3}+0.5z_4^{-0.7}z_5^{-1.6}+0.2z_6^{4.3}z_7^{-1.9}z_8^{8.5}\\&-0.3\left( z_1^{0.5}z_5^{0.5}+z_2^{0.5}z_6^{0.5}+z_3^{0.5}z_7^{0.5}+z_4^{0.5}z_8^{0.5}\right) \\ f_2(\mathbf{z})= & {} 10z_1^{2.3}z_2^{1.7}z_3^{4.5}-z_4^{-2.1}z_5^{0.4}\\ f_3(\mathbf{z})= & {} 0.6z_4^{-2.1}z_5^{0.4}-z_6^{4.5}z_7^{-2.7}z_8^{-0.6}\\ f_4(\mathbf{z})= & {} 6.2z_6^{4.5}z_7^{-2.7}z_8^{-0.6}-z_1^{2.3}z_2^{1.7}z_3^{4.5}\\ f_5(\mathbf{z})= & {} 3.1z_1^{1.6}z_2^{0.4}z_3^{-3.8}-0.3z_4^{5.4}z_5^{1.3}\\ f_6(\mathbf{z})= & {} 3.7z_4^{5.4}z_5^{1.3}-0.1z_6^{-1.1}z_7^{7.3}z_8^{-5.6}\\ f_7(\mathbf{z})= & {} 0.3z_6^{-1.1}z_7^{7.3}z_8^{-5.6}-0.3z_1^{1.6}z_2^{0.4}z_3^{-3.8} \end{aligned}$$

Example 5

Alkylation process optimization (Problem 3 in Dembo [7])

$$\begin{aligned} f_0(\mathbf{z})= & {} 1.715z_1+0.035z_1z_6+4.0565z_3+10z_2+3000-0.063z_3z_5 \\ f_1(\mathbf{z})= & {} 0.00596z_6^2+0.88393z_1^{-1}z_3-0.11756z_6 \\ f_2(\mathbf{z})= & {} 1.1088z_1z_3^{-1}+0.13035z_1z_3^{-1}z_6-0.0066z_1z_3^{-1}z_6^2 \\ f_3(\mathbf{z})= & {} 0.00066z_6^2+0.01724z_5-0.00566z_4-0.01912z_6 \\ f_4(\mathbf{z})= & {} 56.85075z_5^{-1}+1.08702z_5^{-1}z_6+0.32175z_4z_5^{-1}-0.03762z_5^{-1}z_6^2 \\ f_5(\mathbf{z})= & {} 0.0062z_7+2462.312z_2z_3^{-1}z_4^{-1}-25.1256z_2z_3^{-1} \\ f_6(\mathbf{z})= & {} 161.18996z_7^{-1}+5000z_2z_3^{-1}z_7^{-1}-489510z_2z_3^{-1}z_4^{-1}z_7^{-1} \\ f_7(\mathbf{z})= & {} 44.33333z_5^{-1}+0.33z_5^{-1}z_7 \\ f_8(\mathbf{z})= & {} 0.02256z_5-0.0076z_7 \\ f_9(\mathbf{z})= & {} 0.00061z_3-0.0005z_1 \\ f_{10}(\mathbf{z})= & {} 0.81967z_1z_3^{-1}+0.81967z_3^{-1} \\ f_{11}(\mathbf{z})= & {} 24500z_2z_3^{-1}z_4^{-1}-250z_2z_3^{-1} \\ f_{12}(\mathbf{z})= & {} 0.0102z_4+0.00001z_2^{-1}z_3z_4 \\ f_{13}(\mathbf{z})= & {} 0.00006z_1z_6+0.00006z_1 -0.00008z_3 \\ f_{14}(\mathbf{z})= & {} 1.22z_1^{-1}z_3+z_1^{-1}-z_6 \\ \mathbf{z}^L= & {} \{1,1,1,85,90,3,145\} \\ \mathbf{z}^U= & {} \{2000,120,5000,93,95,12,162\} \end{aligned}$$

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Zhan, Y., Zheng, Q.P., Tseng, CL. et al. An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming. J Glob Optim 70, 579–599 (2018). https://doi.org/10.1007/s10898-017-0563-4

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