A chaotic salp swarm algorithm based on quadratic integrate and fire neural model for function optimization

Abstract

Real-world problems generally do not possess mathematical features such as differentiability and convexity and thus require non-traditional approaches to find optimal solutions. SSA is a meta-heuristic optimization algorithm based on the swimming behaviour of salps. Though a novel idea, it suffers from a slow convergence rate to the optimal solution, due to lack of diversity in salp population. In order to improve its performance, chaotic oscillations generated from quadratic integrate and fire model have been augmented to SSA. This improves the balance between exploration and exploitation, generating diversity in the salp population, thus avoiding local entrapment. CSSA has been tested against twenty-two bench mark functions. Its performance has been compared with existing standard optimization algorithms, namely particle swarm optimization, ant–lion optimization and salp swarm algorithm. Statistical tests have been carried out to prove the superiority of chaotic salp swarm algorithm over the other three algorithms. Finally, chaotic SSA is applied on three engineering problems to demonstrate its practicability in real-life applications.

Benchmark functions

In this section, details of the function used in Table 2 have been described.

• Function F1

  • Mathematical expression: $\sum _{i=1}^{n} x_i^2$

  • Lower bound: $-100$

  • Upper bound: 100

  • Dimensions: 30

• Function F2

  • Mathematical expression: $\sum _{i=1}^{n} \left| x_i \right| + \prod _{i=1}^{n} \left| x_i \right|$

  • Upper bound: -10

  • Lower bound: 10

  • Dimensions: 10

• Function F3

  • Mathematical expression: $\sum _{i=1}^{n}\left( \sum _{j=1}^{i} x^{j}\right) ^2$

  • Upper bound: -100

  • Lower bound: 100

  • Dimensions: 10

• Function F4

  • Mathematical expression: $max_i\{ \left| x_i \right| , 1\le i \le n \}$

  • Upper bound: -100

  • Lower bound: 100

  • Dimensions: 10

• Function F5

  • Mathematical expression: $\sum _{i=1}^{n-1}[100(x_{i+1}-x_{i}^2)^2 +(x_i -1)^2 ]$

  • Upper bound: $-30$

  • Lower bound: 30

  • Dimensions: 10

• Function F6

  • Mathematical expression: $\sum _{i=1}^{n}([x_i +0.5])^2$

  • Upper bound: $-100$

  • Lower bound: 100

  • Dimensions: 10

• Function F7

  • Mathematical expression: $\sum _{i=1}^{n} ix_i^4+\hbox {random}[0,1)$

  • Upper bound: $-1.28$

  • Lower bound:1.28

  • Dimensions: 10

• Function F8

  • Mathematical expression: $\sum _{i=1}^{n} ix_i^4+random[0,1)$

  • Upper bound: $-500$

  • Lower bound: 500

  • Dimensions: 10

• Function F9

  • Mathematical expression: $\sum _{i=1}^{n} [x_i^2-10\cos (2\pi x_i)+10]$

  • Upper bound: $-5.12$

  • Lower bound: 5.12

  • Dimensions: 10

• Function F10

  • Mathematical expression: $-20exp\left( {-}0.2\sqrt{\frac{1}{n}\sum _{i{=}1}^{n}x_i^2} \right) -exp\left( \frac{1}{n}\sum _{i=1}^{n}\cos (2\pi x_i) \right) +20 +e$

  • Upper bound: $-32$

  • Lower bound: 32

  • Dimensions: 10

• Function F11

  • Mathematical expression: $\frac{1}{400} \sum _{i=1}^{n}x_i^2 - \prod _{i=1}^{n}\cos \left( \frac{x_i}{\sqrt{i}} \right) +1$

  • Upper bound: $-600$

  • Lower bound: 600

  • Dimensions: 10

• Function F12

  • Mathematical expression: $\frac{\pi }{n} \left\{ \right. 10 \sin (\pi y_1) +\sum _{i=1}^{n-1}(y_i -1)^2 [1 + 10 \sin ^2 (\pi y_{i+1})]+(y_n-1)^2\} + \sum _{i=1}^{n} u(x_i,10,100,4)$

    $y_i=1+\frac{x_i+1}{4}$

    $u(x_i,a,k,m)=\left\{ \begin{array}{ll} k(x_i-a)^m &{} x_i >a\\ 0 &{} -a<x_i<a\\ k(-x_i-a)^m &{} x_i < -a \end{array}\right.$

  • Upper bound: $-50$

  • Lower bound: 50

  • Dimensions: 10

• Function F13

  • Mathematical expression:

    $0.1 \left\{ \sin ^2\left( 3 \pi x_1 \right) \right. + \sum _{i=1}^{n} (x_i-1)^2[1+\sin ^2(3 \pi x_i +1)]+(x_n -1)^2 [1+\sin ^2(2 \pi x_n)]\left. \right\} + \sum _{i=1}^{n} u(x_i,5,100,4)$

    $y_i=1+ \frac{x_i + 1 }{4}$

    $u(x_i,a,k,m)=\left\{ \begin{array}{ll} k(x_i-a)^m &{} x_i >a\\ 0 &{} -a<x_i<a\\ k(-x_i-a)^m &{} x_i < -a \end{array}\right.$

  • Upper bound: $-50$

  • Lower bound: 50

  • Dimensions: 10

• Function F14

  • Pseudocode:

    $\hbox {aS}=[-32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32;,\ldots$

    $-32 -32 -32 -32 -32 -16 -16 -16 -16 -16 0 0 0 0 0 16 16 16 16 16 32 32 32 32 32];$

    $\mathbf{for }\hbox { j}=1:25$

    $\hbox {bS(j)}=\mathbf{sum }(({x}'-\hbox {aS}(:,\hbox {j})).\hat{\,}6)$;

    end

    $\hbox {o}=(1/500+\mathbf{sum }(1./([1:25]+\hbox {bS}))).\hat{\,}(-1)$;

    end

  • Upper bound: $-65.536$

  • Lower bound: 65.536

  • Dimensions: 2

• Function F15

  • Pseudocode:

    $\hbox {aK}=[.1957 .1947 .1735 .16 .0844 .0627 .0456 .0342 .0323 .0235 .0246]$;

    $\hbox {bK}=[.25 .5 1 2 4 6 8 10 12 14 16]$;

    $\hbox {bK}=1./\hbox {bK}$;

    $\hbox {o}=\mathbf{sum }((\hbox {aK}-(({x}(1).^{*}(\hbox {bK}.\hat{\,}2 +{x}(2).^{*}\hbox {bK}))./(\hbox {bK}.\hat{\,}2+{x}(3).^{*} \hbox {bK}+{x}(4)))).\hat{\,}2)$;

    end

  • Upper bound: $-5$

  • Lower bound: 5

  • Dimensions: 4

• Function F16

  • Pseudocode:

    $$\begin{aligned} O&=4*(x(1)^2)-2.1*(x(1)^4)+(x(1)^6)/3\\&\quad +x(1)*x(2)-4*(x(2)^2)+4*(x(2)^4); \end{aligned}$$

  • Upper bound: $-5$

  • Lower bound: 5

  • Dimensions: 4

• Function F17

  • Pseudocode:

    $\hbox {o}=(1+({x}(1)+{x}(2)+1)\hat{\,}2^{*}(19-14^{*}{x}(1) +3^{*}({x}(1)\hat{\,}2)-14^{*}{x}(2)+6^{*}{x}(1)^{*}{x} (2)+3^{*}{x}(2)\hat{\,}2))^{*}\ldots (30+(2^{*}{x}(1)-3^{*}{x}(2))\hat{\,}2^{*}(18-32^{*}{x}(1) +12^{*}({x}(1)\hat{\,}2)+48^{*}{x}(2)-36^{*}{x}(1)^{*}{x} (2)+27^{*}({x}(2)\hat{\,}2)))$;

  • Upper bound: $-2$

  • Lower bound: 2

  • Dimensions: 2

• Function F18

  • Pseudocode:

    $\hbox {aH}=[3 10 30;.1 10 35;3 10 30;.1 10 35]$;

    $\hbox {cH}=[1 1.2 3 3.2]$;

    $\hbox {pH}=[.3689 .117 .2673;.4699 .4387 .747;.1091 .8732 .5547; .03815 .5743 .8828]$;

    $\hbox {o}=0$;

    for $\hbox {i}=1:4$

    $\hbox {o}=\hbox {o}-\hbox {cH(i)}^{*}\hbox {exp}(-(\mathbf{sum }(\hbox {aH}(\hbox {i},:). ^{*}((\hbox {x-pH}(\hbox {i},:)).\hat{\,}2))))$;

    end

  • Upper bound: 0

  • Lower bound: 1

  • Dimensions: 3

• Function F19

  • Pseudocode: $\hbox {aSH}=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3; 8 1 8 1;6 2 6 2;7 3.6 7 3.6]$;

    $\hbox {cSH}=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5]$;

    $\hbox {o}=0$;

    for $\hbox {i}=1:5$

    $\hbox {o}=\hbox {o-((x-aSH(i,:))}^{*}(\hbox {x-aSH}(\hbox {i},:))'+\hbox {cSH}(\hbox {i}))\hat{\,} (-1)$;

    end

  • Upper bound: 0

  • Lower bound: 1

  • Dimensions: 6

• Function F20

  • Pseudocode:

    $\hbox {aH}=[10 3 17 3.5 1.7 8;.05 10 17 .1 8 14;3 3.5 1.7 10 17 8;17 8 .05 10 .1 14]$;

    $\hbox {cH}=[1 1.2 3 3.2]$;

    $\hbox {pH}=[.1312 .1696 .5569 .0124 .8283 .5886;.2329 .4135 .8307 .3736 .1004 .9991$; $\ldots \, .2348 .1415 .3522 .2883 .3047 .6650;.4047 .8828 .8732 .5743 .1091 .0381]$;

    $\hbox {o}=0$;

    for $\hbox {i}=1:4$

    $\hbox {o}=\hbox {o}-\hbox {cH(i)}^{*}\mathbf{exp }(-(\mathbf{sum }(\hbox {aH}(\hbox {i},:).^{*} ((\hbox {x-pH}(\hbox {i},:)).\hat{\,}2))))$;

    end

  • Upper bound: 0

  • Lower bound: 10

  • Dimensions: 4

• Function F21

  • Pseudocode: $\hbox {aSH}=[4 4 4 4;1 1 1 1;8 8 8 8; 6 6 6 6;3 7 3 7; 2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6]$;

    $\hbox {cSH}=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5]$;

    $\hbox {o}=0$;

    for $\hbox {i}=1:7$

    $\hbox {o}=\hbox {o}-((\hbox {x-aSH}(\hbox {i},:))^{*}(\hbox {x-aSH}(\hbox {i},:)) '+\hbox {cSH}(\hbox {i}))\hat{\,}(-1)$;

    end

  • Upper bound: 0

  • Lower bound: 10

  • Dimensions: 4

• Function F22

  • Pseudocode:

    $\hbox {aSH}=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6; 3 7 3 7;2 9 2 9;5 5 3 3; 8 1 8 1;6 2 6 2;7 3.6 7 3.6]$;

    $\hbox {cSH}=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5]$;

    $\hbox {o}=0$;

    for $\hbox {i}=1:10$

    $\hbox {o}=\hbox {o}-((\hbox {x-aSH}(\hbox {i},:))^{*}(\hbox {x-aSH}(\hbox {i},:))' +\hbox {cSH}(\hbox {i}))\hat{\,}(-1)$;

    end

  • Upper bound: 0

  • Lower bound: 10

  • Dimensions: 4