Algorithm 1: The BROMLDE Algorithm Procedure

Input: Objective function:$ℱ$, Search-space limits: ($low,up$), Population size: $N_p$,       Dimension of problem:$D$, Maximal function evaluations: $Ma{x}_{FES},$ Jumping       rate: $J_r$    Output: $Psol:$ Global minimum $Pbest:$ Global minimizer 1  Set the current function evaluations $FES=0$ 2  Set the population bounds randomly generate an initial population $P_0$ 3  for $i=1$ to $N_p$ do    //ROML population initialization (generation g = 0) 4    ${\varphi }_{i,0}=rand(0,1)$ 5    $h_i(FES)={(1+\sqrt[3]{\frac{FES}{Ma{x}_{FES}}})}^{15}$ 6    $C_{i,0}=\{\begin{array}{l}\frac{L_{i,0}^{lower}+L_{i,0}^{upper}}{2}+\frac{L_{i,0}^{lower}+L_{i,0}^{upper}}{2h_i(g)}-\frac{X_{i,0}}{h_i(g)},ifrand(0,1)<0.5,\\ X_{i,0}+{\varphi }_{i,0}(X_{r,0}-X_{i,0}),otherwise,\end{array}$ 7   Check the bounds in the current generation by Equation (17) 8  end 9  Get population $P_g$ by selecting $N_p$ fittest points from $\{P_0\cup P_0^C\}$ 10  $FES=FES+2N_p$ 11  Get $Psol$ and $Pbest$ by Equations (20)–(22) 12  while $FES<Ma{x}_{FES}$ do  //Main loop(generation g > 0) 13   $M_{(i,j),g}=0$   //Generate mutation matrix $(M)$ 14   for $i=1$ to $N_p$ do 15     $u=permute(1:D)$ 16     Generate $\mu$, where $\mu ~\mathit{U}(0,1)$ 17     Generate ${\vartheta }_0,{\vartheta }_0=[3\cdot {\vartheta }_1^3],{\mathit{\vartheta }}_1~\mathit{U}[01],{\vartheta }_0\in \mathit{U}\{1:3\}$ 18     switch ${\vartheta }_0$ do 19       $\mathrm{case}1\mathrm{do}\kappa ={(1-\mu )}^2$; 20       $\mathrm{case}2\mathrm{do}\kappa =2\cdot \mu \cdot (1-\mu );$ 21       $\mathrm{case}3\mathrm{do}\kappa ={\mu }^2;$ 22     end 23     $J=u\left(\lceil 1:\kappa \cdot D\rceil \right)|u=permute\left(1:D\right)$; $M_{(i,J),g}=1$ 24   end 25   Calculate the evolutionary step size $F_g$ by Equation (25) 26   Generate the trial vector $T_g$ by Equations (26) and (27) and control the boundaries of $T_g$ by Equation (28) 27   Update population $P_g$ by Equations (29) and (30) 28   $FES=FES+N_p$ 29   if $rand\le J_r$ then  //ROML population with generation jumping 30    Update the bounds by calculating the smallest and biggest values of all      dimensions in the population $P_g$ 31    for $i=1$ to $N_p$ do 32     ${\varphi }_{i,g}=rand(0,1)$ 33     $h_i(FES)={(1+\sqrt[3]{\frac{FES}{Ma{x}_{FES}}})}^{15}$ 34     $C_{i,g}=\{\begin{array}{l}\frac{L_{i,g}^{lower}+L_{i,g}^{upper}}{2}+\frac{L_{i,g}^{lower}+L_{i,g}^{upper}}{2h_i(g)}-\frac{X_{i,g}}{h_i(g)},ifrand(0,1)<0.5,\\ X_{i,g}+{\varphi }_{i,g}(X_{r,g}-X_{i,g}),otherwise,\end{array}$ 35     Check the bounds in the current generation by Equation (17) 36    end 37    Select $N_p$ fittest individuals from $\{P_g\cup P_g^C\}$ and update the population $P_g$      by Equation (33) 38    $FES=FES+N_p$ 39   end 40   Update $Psol$ and $Pbest$ by Equation (34) 41  end