# Global Optimization for Mixed–Discrete Structural Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Reduces the number of convex terms generated in reformulating the MDSO problems: Compared with the Lin and Tsai [17] method that converts the pure discrete signomial terms to convex terms and then linearizes the inverse transformation functions, this study directly linearizes the pure discrete signomial terms. Therefore, fewer convex terms are introduced in the transformed model;
- Decreases the number of binary variables and constraints required to linearize the inverse transformation functions of discrete variables: Compared with the Lin and Tsai [17] method, this study efficiently linearizes the inverse transformation functions of discrete variables by fewer binary variables and constraints.

## 2. Global Optimization Techniques

**Theorem**

**1.**

**Corollary**

**1.**

## 3. Global Optimization Approach for the MDSO Problems

## 4. Numerical Examples

**Example**

**1.**

^{−5}could be reached after two iterations. The Lin and Tsai [17] method and the proposed method also found identical optimal solution that was closer to the exact globally optimal solution than other existing heuristic methods [25,27,28]. Compared with the data in Table 1, the range reduction techniques significantly decreased the CPU time.

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hedar, A. Studies on Metaheuristics for Continuous Global Optimization Problems Dissertation; Kyoto University: Kyoto, Japan, 2004. [Google Scholar]
- Passy, U.; Wilde, D.J. A geometric programming algorithm for solving chemical equilibrium problems. SIAM J. Appl. Math.
**1968**, 16, 363–373. [Google Scholar] [CrossRef] - Ecker, J.G. Geometric programming: Methods, computations and applications. SIAM Rev.
**1980**, 22, 338–362. [Google Scholar] [CrossRef] - delM Hershenson, M.; Boyd, S.P.; Lee, T.H. Optimal design of a CMOS op-amp via geometric programming. IEEE
**2001**, 20, 1–21. [Google Scholar] [CrossRef] [Green Version] - Feigin, P.D.; Passy, U. The geometric programming dual to the extinction probability problem in simple branching processes. Ann. Prob.
**1981**, 9, 498–503. [Google Scholar] [CrossRef] - Kirschen, P.G.; York, M.A.; Ozturk, B.; Hoburg, W.W. Application of signomial programming to aircraft design. J. Aircr.
**2017**, 55, 965–987. [Google Scholar] [CrossRef] - Shen, P.P.; Zhang, K.C. Global optimization of signomial geometric programming using linear relaxation. Appl. Math. Comput.
**2004**, 150, 99–114. [Google Scholar] [CrossRef] - Floudas, C.A.; Pardalos, P.M. State of the Art in Global Optimization: Computational Methods and Applications; Kluwer Academic Publisher: Boston, MA, USA, 1996. [Google Scholar]
- Maranas, C.D.; Floudas, C.A. Global optimization in generalized geometric programming. Comput. Chem. Eng.
**1997**, 21, 351–369. [Google Scholar] [CrossRef] - Floudas, C.A. Global optimization in design and control of chemical process systems. J. Process. Control
**1999**, 10, 125–134. [Google Scholar] [CrossRef] - Floudas, C.A. Recent advances in global optimization for process synthesis, design and control: Enclosure of all solutions. Comput. Chem. Eng.
**1999**, 23, 963–974. [Google Scholar] [CrossRef] - Floudas, C.A. Deterministic Global Optimization: Theory, Methods and Application; Kluwer Academic Publishers: Boston, CA, USA, 2000. [Google Scholar]
- Tsai, J.F.; Lin, M.H. Global optimization of signomial mixed-integer nonlinear programming problems with free variables. J. Glob. Optim.
**2008**, 42, 39–49. [Google Scholar] [CrossRef] - Lundell, A.; Westerlund, J.; Westerlund, T. Some transformation techniques with applications in global optimization. J. Glob. Optim.
**2009**, 43, 391–405. [Google Scholar] [CrossRef] - Li, H.L.; Lu, H.C. Global optimization for generalized geometric programs with mixed free-sign variables. Oper. Res.
**2009**, 57, 701–713. [Google Scholar] [CrossRef] - Lin, M.H.; Tsai, J.F.; Wang, P.C. Solving engineering optimization problems by a deterministic global approach. Appl. Math. Inf. Sci.
**2012**, 6, 1101–1107. [Google Scholar] - Lin, M.H.; Tsai, J.F. A deterministic global approach for mixed-discrete structural optimization. Eng. Optim.
**2014**, 46, 863–879. [Google Scholar] [CrossRef] - Tsai, J.F.; Lin, M.H. An improved framework for solving NLIPs with signomial terms in the objective or constraints to global optimality. Comput. Chem. Eng.
**2013**, 53, 44–54. [Google Scholar] [CrossRef] - Lu, H.C. A logarithmic method for eliminating binary variables and constraints for the product of free-sign discrete function. Discret. Optim.
**2013**, 10, 11–24. [Google Scholar] [CrossRef] [Green Version] - Vielma, J.P.; Nemhauser, G. Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Prog.
**2011**, 128, 49–72. [Google Scholar] [CrossRef] - Lin, M.H.; Tsai, J.F.; Chang, S.C. A superior linearization method for signomial discrete functions in solving three-dimensional open-dimension rectangular packing problems. Optimization
**2017**, 49, 746–761. [Google Scholar] [CrossRef] - Tsai, J.F.; Lin, M.H.; Peng, L.Y. Finding all global optima of engineering design problems with discrete signomial terms. Eng. Optim.
**2020**, 52, 165–184. [Google Scholar] [CrossRef] - Li, H.L.; Huang, Y.H.; Fang, S.C. A logarithmic method for reducing binary variables and inequality constraints in solving task assignment problems. INFORMS J. Comput.
**2013**, 25, 643–653. [Google Scholar] [CrossRef] - Lu, H.C. Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms. J. Glob. Optim.
**2017**, 68, 95–123. [Google Scholar] [CrossRef] - Erbatur, F.; Hasancebi, O.; Tutuncu, I.; Kilic, H. Optimal design of planar and space structures with genetic algorithms. Comput. Struct.
**2000**, 75, 209–224. [Google Scholar] [CrossRef] - LINGO Release 17.0; LINDO System Inc.: Chicago, IL, USA, 2018.
- Cai, J.; Thierauf, G. Evolution strategies for solving discrete optimization problems. Adv. Eng. Softw.
**1996**, 25, 177–183. [Google Scholar] [CrossRef] - Chen, T.Y.; Chen, H.C. Mixed-discrete structural optimization using a rank-niche evolution strategy. Eng. Optim.
**2009**, 41, 39–58. [Google Scholar] [CrossRef] - Sandgren, E. Nonlinear integer and discrete programming in mechanical design optimization. J. Mech. Des.
**1990**, 112, 223–229. [Google Scholar] [CrossRef]

**Figure 1.**Global optimization approach for the mixed–discrete structural optimization (MDSO) problems.

$\mathit{m}$ | $\mathbf{Solution}\text{}({\mathit{x}}_{1},{\mathit{x}}_{2},{\mathit{x}}_{3},{\mathit{x}}_{4},{\mathit{x}}_{5},{\mathit{x}}_{6},{\mathit{x}}_{7},{\mathit{x}}_{8},{\mathit{x}}_{9},{\mathit{x}}_{10})$ | Objective Value | CPU Time (hh:mm:ss) | Error in Constraint | |
---|---|---|---|---|---|

Lin and Tsai [17] | Proposed Method | ||||

16 | (2.202244, 44.044890, 1.748798, 34.975960, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,850.641469 | 00:02:32 | 00:00:59 | 0.013452 |

32 | (2.203906, 44.078113, 1.749476, 34.989529, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,880.915782 | 00:07:14 | 00:02:21 | 0.003775 |

64 | (2.204395, 44.087898, 1.749733, 34.994651, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,890.838703 | 00:35:19 | 00:06:35 | 0.000923 |

128 | (2.204503, 44.090051, 1.749753, 34.995065, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,892.602486 | 02:57:18 | 00:21:36 | 0.000295 |

**Table 2.**Comparisons of experimental results of Example 1 by different methods with range reduction.

Iteration | $\mathbf{Solution}\text{}({\mathit{x}}_{1},{\mathit{x}}_{2},{\mathit{x}}_{3},{\mathit{x}}_{4},{\mathit{x}}_{5},{\mathit{x}}_{6},{\mathit{x}}_{7},{\mathit{x}}_{8},{\mathit{x}}_{9},{\mathit{x}}_{10})$ | Objective Value | Accumulated CPU Time (hh:mm:ss) | Error in Constraint | |
---|---|---|---|---|---|

Lin and Tsai [17] | Proposed Method | ||||

1 | (2.202244, 44.044890, 1.748798, 34.975960, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,850.641953 | 00:02:42 | 00:00:42 | 0.013452 |

2 | (2.204553, 44.091065, 1.749769, 34.995373, 3.0, 60.0, 3.1, 55.0, 2.6, 50.0) | 63,893.490593 | 00:44:43 | 00:11:36 | 0.000005 |

$\mathbf{Solution}\text{}({\mathit{x}}_{1},{\mathit{x}}_{2},{\mathit{x}}_{3},{\mathit{x}}_{4})$ | Objective Value | CPU Time (hh:mm:ss) | |
---|---|---|---|

Lin and Tsai [17] | Proposed Method | ||

(0.8125, 0.4375, 42, 178) | 6074.99836 | 00:00:48 | 00:00:10 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsai, J.-F.; Lin, M.-H.; Wen, D.-Y.
Global Optimization for Mixed–Discrete Structural Design. *Symmetry* **2020**, *12*, 1529.
https://doi.org/10.3390/sym12091529

**AMA Style**

Tsai J-F, Lin M-H, Wen D-Y.
Global Optimization for Mixed–Discrete Structural Design. *Symmetry*. 2020; 12(9):1529.
https://doi.org/10.3390/sym12091529

**Chicago/Turabian Style**

Tsai, Jung-Fa, Ming-Hua Lin, and Duan-Yi Wen.
2020. "Global Optimization for Mixed–Discrete Structural Design" *Symmetry* 12, no. 9: 1529.
https://doi.org/10.3390/sym12091529