Multi-surrogates and multi-points infill strategy-based global optimization method

Abstract

Surrogate-based global optimization (SBGO) methods are widely used to deal with the computationally expensive black-box optimization problems. In order to reduce the computational source, multiple popular individual surrogates containing polynomial response surface (PRS), radial basis functions (RBF), kriging (KRG) and multiple derived ensemble models are constructed to replace the computationally expensive black-box functions. Moreover, a new multi-points infill strategy is presented to accelerate the optimization. New promising points are located by alternately using a hybrid and adaptive promising sampling (HAPS) method and a multi-start sequential quadratic programming (MSSQP) method. The proposed multi-surrogates and multi-points infill strategy-based global optimization (MSMPIGO) method is examined using eighteen unconstrained optimization problems, six nonlinear constrained engineering problems, and one airfoil design optimization problem. Three basic surrogate PRS, RBF, KRG-based global optimization methods using the similar multi-points infill strategy, PRSMPIGO, RBFMPIGO and KRGMPIGO are both considered as the comparative methods. In comparison with PRSMPIGO, RBFMPIGO, KRGMPIGO and three recently introduced SBGO methods, MSMPIGO shows superior search efficiency and strong robustness in locating the global optima.

Appendix: List of test optimization problems

• (1) Six-hump camel-back function (SC) with n = 2

  $$f\left( {\varvec{x}} \right) = 4x_{1}^{2} - 2.1x_{1}^{4} + {{x_{1}^{6} } \mathord{\left/ {\vphantom {{x_{1}^{6} } 3}} \right. \kern-\nulldelimiterspace} 3} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4} {\kern 1pt} {\kern 1pt} .$$

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• (2) Branin function (BR) with n = 2

  $$\begin{aligned} f\left( {\varvec{x}} \right) =\,& \left[ {x_{2} - 5.1\left( {{{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {2\pi }}} \right. \kern-\nulldelimiterspace} {2\pi }}} \right)^{2} + \left( {{5 \mathord{\left/ {\vphantom {5 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)x_{1} - 6} \right]{\kern 1pt}^{2}\\ & + {10}\left[ {1 - \left( {{1 \mathord{\left/ {\vphantom {1 {8\pi }}} \right. \kern-\nulldelimiterspace} {8\pi }}} \right)} \right]{\text{cos}}x_{1} + 10. \end{aligned}$$

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• (3) Generalized polynomial function (GF) with n = 2

  $$\begin{aligned} f\left( {\varvec{x}} \right) = \,& \left( {1.5 - x_{1} \left( {1 - x_{2} } \right)} \right)^{2} + \left( {2.25 - x_{1} \left( {1 - x_{2}^{2} } \right)} \right)^{2}\\ & + \left( {2.625 - x_{1} \left( {1 - x_{2}^{3} } \right)} \right)^{2} . \end{aligned}$$

  (21)

• (4) Goldstein and price function (GP) with n = 2

  $$\eqalign{ \left( x \right) & = \left[ {1 + {{\left( {{x_1} + {x_2} + 1} \right)}^2}\left( {19 - 14{x_1} + 3x_1^2 - 14{x_2} + 6{x_1}{x_2} + 3x_2^2} \right)} \right] \cr & \quad \times \left[ {30 + {{\left( {2{x_1} - 3{x_2}} \right)}^2}\left( {18 - 32{x_1} + 12x_1^2 + 48{x_2} - 36{x_1}{x_2} + 27x_2^2} \right)} \right] \cr}$$

  (22)

• (5) Shubert function (SE) with n = 2

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \left( {\sum\limits_{i = 1}^{5} {i{\text{cos}}\left( {\left( {i + 1} \right)x_{1} + i} \right)} } \right)\left( {\sum\limits_{i = 1}^{5} {i{\text{cos}}\left( {\left( {i + 1} \right)x_{2} + i} \right)} } \right).$$

  (23)

• (6) Banana function (BA) with n = 2

  $$f\left( {\varvec{x}} \right) = 100\left( {x_{2} - x_{1}^{2} } \right){\kern 1pt} {\kern 1pt}^{2} + \left( {1 - x_{1} } \right){\kern 1pt}^{2} .$$

  (24)

• (7) Himmelblau function (HM) with n = 2

  $$f\left( {\varvec{x}} \right) = \left( {x_{1}^{2} + x_{2} - 11} \right)^{2} + \left( {x_{1} + x_{2}^{2} - 7} \right)^{2} .$$

  (25)

• (8) Cross-IN-TRAY function (CT) with n = 2

  $$f\left( {\varvec{x}} \right) = - 0.0001\left( {\left| {\sin \left( {x_{1} } \right)\sin \left( {x_{2} } \right)\exp \left( {\left| {100 - \frac{{\sqrt {x_{1}^{2} + x_{2}^{2} } }}{\pi }} \right|} \right)} \right| + 1} \right)^{0.1} .$$

  (26)

• (9) Zakharov function (ZK) with n = 2

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \sum\limits_{i = 1}^{2} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{2} {0.5ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{2} {0.5ix_{i} } } \right)^{4} .$$

  (27)

• (10) Trid function (TR6 and TR10) with n = 6, 10

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \sum\limits_{i = 1}^{n} {\left( {x_{i} - 1} \right)} {\kern 1pt}^{2} - \sum\limits_{i = 2}^{n} {x_{i} x_{i - 1} } .$$

  (28)

• (11) Paviani function (PF) with n = 10

  $$f\left( {\varvec{x}} \right) = \sum\limits_{i = 1}^{n} {\left[ {\ln^{2} \left( {x_{i} - 2} \right) + \ln^{2} \left( {10 - x_{i} } \right)} \right]} - \left( {\prod\limits_{i = 1}^{n} {x_{i} } } \right)^{0.2} .$$

  (29)

• (12) Sum squares function (SF12 and SF15) with n = 12, 15

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \sum\limits_{i = 1}^{n} {ix_{i}^{2} } {\kern 1pt} .$$

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• (13) Ellipsoid function (ED12 and ED) with n = 12, 15

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{i} {x_{j}^{2} } } .$$

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• (14) A function of 16 variables (F16) with n = 16

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\alpha_{ij} \left( {x_{i}^{2} + x_{i} + 1} \right)} } \left( {x_{j}^{2} + x_{j} + 1} \right),$$

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  $${a_{ij}}_{\left( {row1 - 8} \right)} = \left[ \begin{array}{llllllllllllllll} 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{array} \right]$$
  $${a_{ij}}_{\left( {row 9 - 16} \right)} = \left[ \begin{array}{llllllllllllllll} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right]$$

• (15) Sphere function (SP) with n = 20

  $$f\left( {\varvec{x}} \right) = {\kern 1pt} \sum\limits_{i = 1}^{n} {x_{i}^{2} } {\kern 1pt} .$$

  (33)

• (16) Tension/compression spring design (TSD) with n = 3

This problem aims to minimize the weight of a tension/compression subject to constraints on minimum deflection, shear stress and surge frequency, limits on outside diameter and side constraints.

$$\begin{gathered} \min {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} f\left( {\varvec{x}} \right) = x_{1}^{2} x_{2} \left( {x_{3} + 2} \right) \hfill \\ s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{1} \left( {\varvec{x}} \right) = 1 - \frac{{x_{2}^{3} x_{3} }}{{71785x_{1}^{4} }} \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{2} \left( {\varvec{x}} \right) = \frac{{4x_{2}^{2} - x_{1} x_{2} }}{{12566x_{1}^{3} \left( {x_{2} - x_{1} } \right)}} + \frac{1}{{5108x_{1}^{2} }} - 1 \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{3} \left( {\varvec{x}} \right) = 1 - \frac{{140.45x_{1} }}{{x_{2}^{2} x_{3} }} \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{4} \left( {\varvec{x}} \right) = \frac{{x_{1} + x_{2} }}{1.5} - 1 \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.05 \le x_{1} \le 2;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.25 \le x_{2} \le 1.3;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2 \le x_{3} \le 15. \hfill \\ \end{gathered}$$

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• (17) I-beam design (IBD) with n = 4

This problem aims to minimize the vertical deflection of an I-beam and meanwhile satisfies the cross-section area and stress constraints under given loads.

$$\eqalign{ \min f\left( \varvec{x} \right) & = \frac{{5000}}{{\frac{1}{{12}}{x_3}{{\left( {{x_1} - 2{x_4}} \right)}^3} + \frac{1}{6}{x_2}x_4^3 + 2{x_2}{x_4}{{\left( {\frac{{{x_1} - {x_4}}}{2}} \right)}^2}}} \cr s.t.\,{g_1}\left( \varvec{x} \right) & = 2{x_2}{x_4} + {x_3}\left( {{x_1} - 2{x_4}} \right) - 300 \leq 0 \cr {g_2}\left( \varvec{x} \right) & = \frac{{180000{x_1}}}{{{x_3}{{\left( {{x_1} - 2{x_4}} \right)}^3} + 2{x_2}{x_4}\left( {4x_4^2 + 3{x_1}\left( {{x_1} - 2{x_4}} \right)} \right)}} + \frac{{15000{x_2}}}{{\left( {{x_1} - 2{x_4}} \right)x_3^3 + 2{x_4}x_2^3}} \cr & \quad - 6 \leq 010 \leq {x_1} \leq 80;\;10 \leq {x_2} \leq 50;\;0.9 \leq {x_3},{x_4} \leq 5. \cr}$$

(35)

• (18) Welded beam design (WBD) with n = 4

This problem aims to minimize the cost and meanwhile meets the constraints on shear stress, bending stress in the beam, buckling load on the bar, end deflection of the beam and side constraints.

$$\eqalign{ & \min f\left( \varvec{x} \right) = 1.10471x_1^2{x_2} + 0.04811{x_3}{x_4}\left( {14 + {x_2}} \right) \cr & s.t. {g_1}\left( \varvec{x} \right) = \tau \left( \varvec{x} \right) - {\tau _{\max }} \leq 0 \cr & {g_2}\left( \varvec{x} \right) = \sigma \left( x \right) - {\sigma _{\max }} \leq 0 \cr & {g_3}\left( \varvec{x} \right) = {x_1} - {x_4} \leq 0 \cr & {g_4}\left( \varvec{x} \right) = 0.10471x_1^2 + 0.04811{x_3}{x_4}\left( {14 + {x_2}} \right) - 5 \leq 0 \cr & {g_5}\left( \varvec{x} \right) = 0.125 - {x_1} \leq 0 \cr & {g_6}\left( \varvec{x} \right) = \delta \left( x \right) - {\delta _{\max }} \leq 0 \cr & {g_7}\left( \varvec{x} \right) = P - {P_c}\left( \varvec{x} \right) \leq 0 \cr & where \cr & \tau \left( \varvec{x} \right) = \sqrt {{{\left( {\tau '} \right)}^2} + 2\tau '\tau ''\frac{{{x_2}}}{{2R}} + {{\left( {\tau ''} \right)}^2}} , \tau ' = \frac{P}{{\sqrt 2 {x_1}{x_2}}}, \tau '' = \frac{{MR}}{J}, \cr & M = P\left( {L + \frac{{{x_2}}}{2}} \right), R = \sqrt {\frac{{x_2^2}}{4} + {{\left( {\frac{{{x_1} + {x_3}}}{2}} \right)}^2}} , J = 2\sqrt 2 {x_1}{x_2}\left( {\frac{{x_2^2}}{{12}} + {{\left( {\frac{{{x_1} + {x_3}}}{2}} \right)}^2}} \right), \cr & \sigma \left( \varvec{x} \right) = \frac{{6PL}}{{x_3^2{x_4}}}, \delta \left( \varvec{x} \right) = \frac{{4P{L^3}}}{{Ex_3^3{x_4}}}, {P_c}\left(\varvec{x} \right) = \frac{{4.013E{x_3}x_4^3}}{{6{L^2}}}\left( {1 - \frac{{{x_3}}}{{2L}}\sqrt {\frac{E}{{4G}}} } \right), \cr & G = 12*{10^6}psi, P = 6000 lb, L = 14 in, {\delta _{\max }} = 0.25 in, E = 30*{10^6}psi, \cr & {\tau _{\max }} = 13600 psi, {\sigma _{\max }} = 30000 psi, 0.1 \leq {x_1}, {x_4} \leq 2; 0.1 \leq {x_2}, {x_3} \leq 10 \cr}$$

(36)

• (19) Pressure vessel design (PVD) with n = 4

This problem is designed to minimize the fabrication cost covering the cost of materials, forming, and welding of the pressure vessel. Four design variables are thickness of the pressure vessel, thickness of the head, inner radius of the pressure vessel, and length of the vessel.

$$\begin{gathered} \min {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} f\left( {\varvec{x}} \right) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{3}^{2} + 3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3} \hfill \\ s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{1} \left( {\varvec{x}} \right) = - x_{1} + 0.0193x_{3} \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{2} \left( {\varvec{x}} \right) = - x_{2} + 0.00954x_{3} \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{3} \left( {\varvec{x}} \right) = - \pi x_{3}^{2} x_{4} - {4 \mathord{\left/ {\vphantom {4 {3\pi x_{3}^{3} + }}} \right. \kern-\nulldelimiterspace} {3\pi x_{3}^{3} + }}1296000 \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} g_{4} \left( {\varvec{x}} \right) = x_{4} - 240 \le 0 \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.0625 \le x_{1} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} x_{2} \le 6.1875;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 10 \le x_{3} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} x_{4} \le 200. \hfill \\ \end{gathered}$$

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• (20) Himmelblau’ s nonlinear optimization problem (HIM) with n = 5

This problem has five design variables, six nonlinear inequality constraints and ten boundary conditions.

$$\begin{gathered} \min f\left( {\varvec{x}} \right) = 5.3578547x_{3}^{2} + 0.8356891x_{1} x_{5} + 37.293239x_{1} - 40792.141 \hfill \\ s.t. \,g_{1} \left( {\varvec{x}} \right) = 85.334407 + 0.0056858x_{2} x_{5} + 0.00026x_{1} x_{4} - 0.0022053x_{3} x_{5} \hfill \\ g_{2} \left( {\varvec{x}} \right) = 80.51249 + 0.0071317x_{2} x_{5} + 0.0029955x_{1} x_{2} + 0.0022053x_{3}^{2} \hfill \\ g_{3} \left( {\varvec{x}} \right) = 9.300961 + 0.0047026x_{3} x_{5} + 0.0012547x_{1} x_{3} + 0.0019085x_{3} x_{4} \hfill \\ 0 \le g_{1} \left( {\varvec{x}} \right) \le 92, 90 \le g_{2} \left( {\varvec{x}} \right) \le 110, 20 \le g_{3} \left( {\varvec{x}} \right) \le 25 \hfill \\ 78 \le x_{1} \le 102; 33 \le x_{2} \le 45; 27 \le x_{3} , x_{4} , x_{5} \le 45. \hfill \\ \end{gathered}$$

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• (21) Speed reducer design (SRD) with n = 7

This problem is designed to minimize the total weight of the speed reducer. It has eleven constraints involving the limits on the bending stress of the gear teeth, surface stress and transverse deflections of shafts.

$$\eqalign{ & \min {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} f\left( \varvec{x} \right) = 0.7854{x_1}x_2^2\left( {3.3333x_3^2 + 14.9334{x_3} - 43.0934} \right) - 1.508{x_1}\left( {x_6^2 + x_7^2} \right) \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 7.4777\left( {x_6^3 + x_7^3} \right) + 0.7854\left( {{x_4}x_6^2 + {x_5}x_7^2} \right) \cr & s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_1}\left(\varvec{x} \right) = \frac{{27}}{{{x_1}x_2^2{x_3}}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_6}\left(\varvec{x} \right) = \frac{{{{\left[ {{{\left( {{{745{x_5}} \mathord{\left/ {\vphantom {{745{x_5}} {\left( {{x_2}{x_3}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{x_2}{x_3}} \right)}}} \right)}^2} + 157.5 \times {{10}^6}} \right]}^{0.5}}}}{{85x_7^3}} - 1 \leq 0, \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_2}\left(\varvec{x} \right) = \frac{{397.5}}{{{x_1}x_2^2x_3^2}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_7}\left(\varvec{x} \right) = \frac{{{x_2}{x_3}}}{{40}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_8}\left(\varvec{x} \right) = \frac{{5{x_2}}}{{{x_1}}} - 1 \leq 0 \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_3}\left(\varvec{x} \right) = \frac{{1.93x_4^3}}{{{x_2}{x_3}x_6^4}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_9}\left(\varvec{x} \right) = \frac{{{x_1}}}{{12{x_2}}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_{10}}\left(\varvec{x} \right) = \frac{{1.5{x_6} + 1.9}}{{{x_4}}} - 1 \leq 0 \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_4}\left(\varvec{x} \right) = \frac{{1.93x_5^3}}{{{x_2}{x_3}x_7^4}} - 1 \leq 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_{11}}\left(\varvec{x} \right){\text{ = }}\frac{{1.1{x_7} + 1.9}}{{{x_5}}} - 1 \leq 0 \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g_5}\left(\varvec{x} \right) = \frac{{{{\left[ {{{\left( {{{745{x_4}} \mathord{\left/ {\vphantom {{745{x_4}} {\left( {{x_2}{x_3}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{x_2}{x_3}} \right)}}} \right)}^2} + 16.9 \times {{10}^6}} \right]}^{0.5}}}}{{110x_6^3}} - 1 \leq 0 \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2.6 \leq {x_1} \leq 3.6;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.7 \leq {x_2} \leq 0.8;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 17 \leq {x_3} \leq 28 \cr & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 7.3 \leq {x_4},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {x_5} \leq 8.3;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2.9 \leq {x_6} \leq 3.9;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 5 \leq {x_7} \leq 5.5 \cr}.$$

(39)