Nash game based efficient global optimization for large-scale design problems

Abstract

A novel Nash-EGO algorithm is presented to extend the usage of efficient global optimization (EGO) into large-scale optimizations by coupling with Nash game strategy. In our Nash-EGO, the large-scale design variables are split into several subsets by adopting Nash variable territory splitting, and the EGO optimizer acts as a player of specific Nash game. All the EGO players are coupled with each other and assigned to optimize their own subsets synchronously in parallel to produce the corresponding approximate optimal subsets. Doing in this way, the performance of EGO players could be expected to keep at a high level due to the fact that EGO players now take care of only their own small-scale subsets instead of facing the large-scale problem directly. A set of typical cases with a small number of variables are firstly selected to validate the performance of each EGO player mentioned. Then, the Nash-EGO proposed is tested by representative functions with a scale up to 30 design variables. Finally, more challenge cases with 90 design variables are constructed and investigated to mimic the real large-scale optimizations. It can be learned from the tests that, with respect to conventional EGO the present algorithm can always find near optimal solutions, which are more close to the theoretical values, and are achieved, moreover, less CPU time-consuming, up to hundreds times faster. All cases with 30 or 90 design variables have similar efficient performances, which indicates the present algorithm has the potential to cope with real large-scale optimizations.

Appendices

Formulas of test functions in Sect. 3.1

(1) Branin function

$$\begin{aligned} f_1 =\left( {x_2 -\frac{5.1}{4\pi ^{2}}x_1^2 +\frac{5}{\pi }x_1 -6} \right) ^{2}+10\left( {1-\frac{1}{8\pi }}\right) \cos x_1 +10 \end{aligned}$$

(A. 1)

where $-5\le x_1 \le 10$, $0\le x_2 \le 15$, its minimum is 0.3979.

(2) Hartman 3 function

$$\begin{aligned} f_2 =-\mathop \sum \limits _{i=1}^4 c_i exp\left[ -\mathop \sum \limits _{j=1}^3 a_{ij} (x_j -p_{ij} )^{2}\right] \end{aligned}$$

(A. 2)

where $a_{ij} =\left[ {{\begin{array}{lll} 3 &{}1&{} {30} \\ {0.1}&{} {10}&{} {35} \\ 3&{} {10}&{} {30} \\ {0.1}&{} {10}&{} {35} \\ \end{array} }} \right]$, $c_{ij} =\left[ {{\begin{array}{l} 1 \\ {1.2} \\ 3 \\ {3.2} \\ \end{array} }} \right]$, $p_{ij} =\left[ {{\begin{array}{lll} {0.3689}&{} {0.117}&{} {0.2673} \\ {0.4699}&{} {0.4389}&{} {0.747} \\ {0.1091}&{} {0.8732}&{} {0.5547} \\ {0.03815}&{} {0.5743}&{} {0.8828} \\ \end{array} }} \right]$, $0\le x_i \le 1$, $i=1, 2, 3$, its minimum is $-3.8628$.

(3) Hartman 6 function

$$\begin{aligned} f_3 =-\mathop \sum \limits _{i=1}^4 c_i exp\left[ -\mathop \sum \limits _{j=1}^6 a_{ij} (x_j -p_{ij} )^{2}\right] \end{aligned}$$

(A. 3)

where $a_{ij} =\left[ {\begin{array}{llllll} {10}&{} 3&{} {17} &{}{3.5}&{} {1.7}&{} 8 \\ {0.05}&{} {10}&{} {17} &{} {0.1}&{} 8&{} {14} \\ 3&{} {3.5}&{} {1.7} &{} {10}&{} {17}&{} 8 \\ {17}&{} 8&{} {0.05} &{} 0&{} {0.}&{} {14} \\ \end{array} } \right]$, $c_{ij} =\left[ {{\begin{array}{l} 1 \\ {1.2} \\ 3 \\ {3.2} \\ \end{array} }} \right]$, $0\le x_i \le 1$, $i=1,2,\ldots ,6$,

$p_{ij} =\left[ {{\begin{array}{llllll} {0.1312}&{} {0.1696}&{} {0.5569} &{} {0.0124}&{} {0.8283}&{} {0.5886} \\ {0.2329}&{} {0.4135}&{} {0.8307} &{} {0.3736}&{} {0.1004}&{} {0.9991} \\ {0.2348}&{} {0.1451}&{} {0.3522} &{} {0.2883}&{} {0.3047}&{} {0.6650} \\ {0.4047}&{} {0.8828}&{} {0.8732} &{} {0.5743}&{} {0.1091}&{} {0.0381} \\ \end{array} }} \right] ,$ its minimum is $-3.3224$.

Formulas of test functions in Sect. 3.2

(1) Sphere function

$$\begin{aligned} f_4 =\sum \limits _{i=1}^D {x_i^2 } \end{aligned}$$

(A. 4)

where D is the number of variables, $-100\le x_i \le 100$, its minimum is 0.

(2) Elliptic function

$$\begin{aligned} f_5 =\sum \limits _{i=1}^D {({10^{6}})^{\left[ {{({i-1} )}/{({D-1})}} \right] }x_i^2 } \end{aligned}$$

(A. 5)

where D is the number of variables, $-10\le x_i \le 10$, its minimum is 0.

(3) Ackley function

$$\begin{aligned} f_6 =-20\exp \left( {-0.2\sqrt{D^{-1}\sum _{i=1}^D {x_i^2 } }} \right) -\exp \left( {D^{-1}\sum _{i=1}^D {\cos 2\pi x_i } } \right) +20+e \end{aligned}$$

(A. 6)

where D is the number of variables, $-32\le x_i \le 32$, its minimum is 0.

(4) Rastrigin function

$$\begin{aligned} f_7 =\sum \limits _{i=1}^D {\left[ {x_i^2 -10\cos ({2\pi x_i } )+10} \right] } \end{aligned}$$

(A. 7)

where D is the number of variables, $-5.12\le x_i \le 5.12$, its minimum is 0.

(5) Shifted and rotated Ackley function

$$\begin{aligned} f_6 =-20\exp \left( {-0.2\sqrt{D^{-1}\sum _{i=1}^D {z_i^2 } }} \right) -\exp \left( {D^{-1}\sum _{i=1}^D {\cos 2\pi z_i } } \right) +20+e+f_{bias}\nonumber \\ \end{aligned}$$

(A. 8)

where $\mathbf{z}=\mathbf{M}({\mathbf{x-o}})$, $\mathbf{z}=({z_1 , z_2 , \ldots , z_D })$, $x=({x_1 , x_2 , \ldots , x_D })$, D is the number of variables, $\mathbf{M}$ is the linear transformation matrix, and $\mathbf{o}$ is the shifted global optimum, their values can be refer to Ref. [42, 43]. $-32\le x_i \le 32$, its minimum is $f_{bias}$, in this paper, it is set as $f_{bias} =-140$.