Multi-Verse Optimizer: a nature-inspired algorithm for global optimization

Abstract

This paper proposes a novel nature-inspired algorithm called Multi-Verse Optimizer (MVO). The main inspirations of this algorithm are based on three concepts in cosmology: white hole, black hole, and wormhole. The mathematical models of these three concepts are developed to perform exploration, exploitation, and local search, respectively. The MVO algorithm is first benchmarked on 19 challenging test problems. It is then applied to five real engineering problems to further confirm its performance. To validate the results, MVO is compared with four well-known algorithms: Grey Wolf Optimizer, Particle Swarm Optimization, Genetic Algorithm, and Gravitational Search Algorithm. The results prove that the proposed algorithm is able to provide very competitive results and outperforms the best algorithms in the literature on the majority of the test beds. The results of the real case studies also demonstrate the potential of MVO in solving real problems with unknown search spaces. Note that the source codes of the proposed MVO algorithm are publicly available at http://www.alimirjalili.com/MVO.html.

Appendices

Appendix 1

Appendix 2

2.1 Welded beam design problem

$$\begin{array}{ll} {\text{Consider}} & {\vec{x} = \left[{x_{1} \,x_{2} \,x_{3} \,x_{4}} \right] = \left[{h \,l\, t\, b} \right],} \\ {\text{Minimize}} & {f\left({\vec{x}} \right) = 1.10471x_{1}^{2} x_{2}} \\ & {\quad +\,0.04811x_{3} x_{4} \left( {14.0 + x_{2} } \right),} \\ {\text{Subject to}} & {g_{1} \left( {\vec{x}} \right) = \tau \left( {\vec{x}} \right) - \tau_{\hbox{max} } \le 0,} \\ & {g_{2} \left( {\vec{x}} \right) = \sigma \left( {\vec{x}} \right) - \sigma_{\hbox{max} } \le 0,} \\ & {g_{3} \left( {\vec{x}} \right) = \delta \left( {\vec{x}} \right) - \delta_{\hbox{max} } \le 0} \\ & {g_{4} \left( {\vec{x}} \right) = x_{1} - x_{4} \le 0,} \\ & {g_{5} \left( {\vec{x}} \right) = P - P_{c} (\vec{x}) \le 0,} \\ & {g_{6} \left( {\vec{x}} \right) = 0.125 - x_{1} \le 0} \\ & {g_{7} \left( {\vec{x}} \right) = 1.10471x_{1}^{2}} \\ & {\quad +\,0.04811x_{3} x_{4} \left( {14.0 + x_{2} } \right) - 5.0 \le 0} \\ {\text{Variable range}} & {0.1 \le x_{1} \le 2,} \\ & {0.1 \le x_{2} \le 10,}\\ & {0.1 \le x_{3} \le 10,} \\ & {0.1 \le x_{4} \le 2} \\ {\text{where}} & {\tau \left( {\vec{x}} \right) = \sqrt {(\tau^{{\prime }} )^{2} + 2\tau^{\prime} \tau^{\prime \prime} \frac{{x_{2} }}{2R} + (\tau^{\prime \prime} )^{2} } ,} \\ & {\tau^{{\prime }} = \frac{P}{{\sqrt{2} x_{1} x_{2} }}, \quad \tau^{{\prime \prime }} = \frac{\text{MR}}{J},}\\ & {M = P\left( {L + \frac{{x_{2} }}{2}} \right), }\hfill \\ {} \hfill & {R = \sqrt {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } ,} \hfill \\ {} \hfill & {J = 2\left\{ {\sqrt{2} x_{1} x_{2} \left[ {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } \right]} \right\},} \hfill \\ {} \hfill & {\sigma \left( {\vec{x}} \right) = \frac{{6{\text{PL}}}}{{x_{4} x_{3}^{2} }},\quad \delta \left( {\vec{x}} \right) = \frac{{6{\text{PL}}^{3} }}{{Ex_{3}^{2} x_{4} }}} \hfill \\ {} \hfill & {P_{c} \left( { \vec{x}} \right) = \frac{{4.013E\sqrt {\frac{{x_{3}^{2} x_{4}^{6} }}{36}} }}{{L^{2} }}\left( {1 - \frac{{x_{3} }}{2L}\sqrt{\frac{E}{4G}} } \right),} \hfill \\ {} \hfill & {P = 6000 \;{\text{lb}},\,\, L = 14\; {\text{in}} ., \,\,\delta_{\hbox{max} } = 0.25\; {\text{in}} .,} \\ & {E = 30 \times 1^{6} \; {\text{psi}}, \quad G = 12 \times 10^{6} \;{\text{psi}},} \\ & {\tau_{ \hbox{max} } = 13600\;{\text{psi}},\;\;\sigma_{ \hbox{max} } = 30000\;{\text{psi}}} \hfill \\ \end{array}$$

2.2 Gear train design problem

$$\begin{array}{*{20}l} {\text{Consider}} \hfill & {\vec{x} = \left[ {x_{1} \, x_{2} \, x_{3} \,x_{4} } \right] = \left[ {n_{A} \,n_{B} \,n_{C} \,n_{D} } \right],} \hfill \\ {\text{Minimize}} \hfill & {f\left( {\vec{x}} \right) = \left( {\frac{1}{6.931} - \frac{{x_{3} x_{2} }}{{x_{1} x_{4} }}} \right)^{2} ,} \hfill \\ {\text{Variable range}} \hfill & {12 \le x_{1} ,x_{2} ,x_{3} , x_{4} \le 60,} \hfill \\ \end{array}$$

2.3 Three-bar truss design problem

$$\begin{array}{ll} {\text{Consider}} \hfill & {\vec{x} = \left[ {x_{1} \,x_{2} } \right] = \left[ {A_{1} \,A_{2} } \right],} \hfill \\ {\text{Minimize}} \hfill & { f\left( {\vec{x}} \right) = \left( {2\sqrt{2} x_{1} + x_{2} } \right)*l,} \hfill \\ {\text{Subject to }} \hfill & {g_{1} \left( {\vec{x}} \right) = \frac{{\sqrt{2} x_{1} + x_{2} }}{{\sqrt{2} x_{1}^{2} + 2x_{1} x_{2} }}P - \sigma \le 0,} \hfill \\ {} \hfill & { g_{2} \left( {\vec{x}} \right) = \frac{{x_{2} }}{{\sqrt{2} x_{1}^{2} + 2x_{1} x_{2} }}P - \sigma \le 0,} \hfill \\ {} \hfill & { g_{3} \left( {\vec{x}} \right) = \frac{1}{{\sqrt{2} x_{2} + x_{1} }}P - \sigma \le 0,} \hfill \\ {\text{Variable range }} \hfill & {0 \le x_{1} ,x_{2} \le 1,} \hfill \\ {\text{where }} \hfill & l = 100 \,\,{\text{cm}}, P = 2\,\,{\text{KN}}/{\text{cm}}^{2} ,\\ & \sigma = 2\,\,{\text{KN}}/{\text{cm}}^{2} \hfill \\ \end{array}$$

2.4 Pressure vessel design problem

$$\begin{array}{ll} {\text{Consider }} \hfill & {\vec{x} = \left[ {x_{1} \,x_{2} \,x_{3} \,x_{4} } \right] = \left[ {T_{\text{s}} \,T_{\text{h}} \,R \,L} \right],} \hfill \\ {\text{Minimize }} \hfill & {f\left( {\vec{x}} \right) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{3}^{2}} \\ &\quad {+\,3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3},} \hfill \\ {\text{Subject to}} \hfill & { g_{1} \left( {\vec{x}} \right) = - x_{1} + 0.0193x_{3} \le 0,} \hfill \\ {} \hfill & { g_{2} \left( {\vec{x}} \right) = - x_{3} + 0.00954x_{3} \le 0,} \hfill \\ {} \hfill & {g_{3} \left( {\vec{x}} \right) = - \pi x_{3}^{2} x_{4} - \frac{4}{3}\pi x_{3}^{3}} \\ &\quad {+\,1296000 \le 0,} \hfill \\ {} \hfill & { g_{4} \left( {\vec{x}} \right) = x_{4} - 240 \le 0,} \hfill \\ {\text{Variable range }} \hfill & {0 \le x_{1} \le 99,} \hfill \\ {} \hfill & { 0 \le x_{2} \le 99, } \hfill \\ {} \hfill & { 10 \le x_{3} \le 200, } \hfill \\ {} \hfill & { 10 \le x_{4} \le 200} \hfill \\ \end{array}$$

2.5 Cantilever beam design

$$\begin{array}{*{20}l} {\text{Consider}} \hfill & {\vec{x} = \left[ {x_{1} x_{2} x_{3} x_{4} x_{5} } \right]} \hfill \\ {\text{Minimize}} \hfill & {f\left( {\vec{x}} \right) = 0.6224\left( {x_{1} + x_{2} + x_{3} + x_{4} + x_{5} } \right),} \hfill \\ {\text{Subject to }} \hfill & { g\left( {\vec{x}} \right) = \frac{61}{{x_{1}^{3} }} + \frac{27}{{x_{2}^{3} }} + \frac{19}{{x_{3}^{3} }} + \frac{7}{{x_{4}^{3} }} + \frac{1}{{x_{5}^{3} }} - 1 \le 0,} \hfill \\ {\text{Variable range }} \hfill & {0.01 \le x_{1} , x_{2} , x_{3} , x_{4} , x_{5} \le 100,} \hfill \\ \end{array}$$