An advanced initialization technique for metaheuristic optimization: a fusion of Latin hypercube sampling and evolutionary behaviors

Abstract

Many new metaheuristic algorithms prioritize their search strategy phase, often neglecting equally critical stages like initialization. Latin hypercube sampling (LHS) is one technique that stands out in this context. LHS selects representative samples through permutations in a multidimensional space, effectively preventing the clustering of points in specific areas. However, its limitations become apparent in high-dimensional problems, as it fails to provide crucial search space information and struggles to determine samples across all dimensions. Addressing these challenges, this paper introduces an innovative population initialization approach, combining the strengths of LHS with evolutionary behaviors. This technique is divided into two main sections: spatial and quality. The spatial section divides the search space into equal intervals across each dimension to establish initial solutions. Meanwhile, the quality section employs evolutionary strategies like mutation and crossover. These strategies serve a dual purpose: they explore the search space thoroughly and refine solutions, bringing them closer to the objective function. To validate the effectiveness of this method, these principles have been integrated it into the classic Differential Evolution algorithm. We conducted extensive tests using 30 representative benchmark functions to assess its performance. The experimental results are encouraging; our methodology not only speeds up convergence but also enhances solution quality, outperforming other similar techniques.

Appendix

	Name	Minimum	S	D	Function
${f(x)}_{1}$	Ackley	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-30, 30]}^{d}$	30	$f\left(x\right)=-20{\text{exp}}\left(-0.2\sqrt{\frac{1}{d}\sum\limits_{i=d}^{d}{x}_{i}^{2}}\right)-{\text{exp}}\left(\frac{1}{d}\sum\limits_{i=1}^{d}{\text{cos}}\left(2\pi {x}_{i}\right)\right)+{\text{exp}}\left(1\right)+20$
${f(x)}_{2}$	Dixon-Price	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}={2}^{-\frac{{2}^{i}-2}{{2}^{i}}}\mathrm{ for} i=1,\dots ,n$	${[-10, 10]}^{d}$	30	$f\left(x\right)={({x}_{1}-1)}^{2}+\sum\limits_{i=2}^{d}i{(2{x}_{i}^{2}-{x}_{i-1})}^{2}$
${f(x)}_{3}$	Griewank	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-600, 600]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}\frac{{x}_{i}^{2}}{4000}-\prod_{i=1}^{d}{\text{cos}}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$
${f(x)}_{4}$	Infinity	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-1, 1]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{6}(sen\left({x}_{i}\right)+2)$
${f(x)}_{5}$	Levy	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(1,\dots , 1\right)$	${[-10, 10]}^{d}$	30	$f\left(x\right)={sin}^{2}\left(\pi {\omega }_{1}\right)+\sum\limits_{i}^{d-1}{\left({\omega }_{1}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {\omega }_{i}+1\right)\right]+{\left({\omega }_{d}-1\right)}^{2}\left[1+\right]{sin}^{2}\left(2\pi {\omega }_{d}\right) whre {\omega }_{i}=1+\frac{{x}_{i}-1}{4}$
${f(x)}_{6}$	Mishra 1	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=2; {\mathbf{x}}^{\mathbf{*}}=\left(1,\dots , 1\right)$	${[0, 1]}^{d}$	30	$f\left(x\right)={(1+(d-\sum\limits_{i=1}^{d-1}{x}_{i}))}^{d-\sum\limits_{i=1}^{d-1}{x}_{i}}$
${f(x)}_{7}$	Mishra 2	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=2; {\mathbf{x}}^{\mathbf{*}}=\left(1,\dots , 1\right)$	${[0, 1]}^{d}$	30	$f\left(x\right)={\left(1+\left(d-\sum\limits_{i=1}^{d-1}\frac{{x}_{i}+{x}_{i+1}}{2}\right)\right)}^{d-\sum\limits_{i=1}^{d-1}(\frac{{x}_{i}+{x}_{i+1}}{2})}$
${f(x)}_{8}$	Mishra 11	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-10, 10]}^{d}$	30	$f\left(x\right)={\left[\frac{1}{d}\sum\limits_{i=1}^{d}\left|{x}_{i}\right|-{\left(\prod_{i=1}^{d}|{x}_{i}|\right)}^\frac{1}{d}\right]}^{2}$
${f(x)}_{9}$	MultiModal	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-10, 10]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}|{x}_{i}|\prod_{i=1}^{d}|{x}_{i}|$
${f(x)}_{10}$	Penalty 1	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(-1,\dots , -1\right)$	${[-50, 50]}^{d}$	30	$f\left(x\right)=\frac{\pi }{30}\left(10{sin}^{2}\left(\pi {y}_{1}\right)+\sum\limits_{i=1}^{d-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{i}-1\right)}^{2}\right)+\sum\limits_{i=1}^{d}u\left({x}_{i},\mathrm{10,100,4}\right) {y}_{i}=1+\frac{{x}_{i}+1}{4}, u({x}_{i},a,k,m)\left\{\begin{array}{c}\begin{array}{cc}k{({x}_{i}-a)}^{m}& ,{x}_{i}>a\end{array}\\ \begin{array}{cc}0& ,-a\le {x}_{i}\le a\end{array}\\ \begin{array}{cc}k{(-{x}_{i}-a)}^{m}& ,{x}_{i}<-a\end{array}\end{array}\right.$
${f(x)}_{11}$	Penalty 2	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(1,\dots , 1\right)$	${[-50, 50]}^{d}$	30	$f\left(x\right)=0.1\left({(sin\left(3\pi {x}_{1}\right))}^{2}+\sum\limits_{i=1}^{d-1}{\left({x}_{i}-1\right)}^{2}\left[1+{sin}^{2}\left(3\pi {x}_{i+1}\right)\right]+\left[{\left({x}_{i}-1\right)}^{2}{(sin\left(2\pi {x}_{i}\right))}^{2}\right]\right)+\sum\limits_{i=1}^{d}u\left({x}_{i},\mathrm{5,100,4}\right) u({x}_{i},a,k,m)\left\{\begin{array}{c}\begin{array}{cc}k{({x}_{i}-a)}^{m}& ,{x}_{i}>a\end{array}\\ \begin{array}{cc}0& ,-a\le {x}_{i}\le a\end{array}\\ \begin{array}{cc}k{(-{x}_{i}-a)}^{m}& ,{x}_{i}<-a\end{array}\end{array}\right.$
${f(x)}_{12}$	Perm 1	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(1, 2,\dots , n\right)$	${[-d, d]}^{d}$	30	$f\left(x\right)=\sum\limits_{k=1}^{d}{\left[\sum\limits_{i=1}^{d}({i}^{k}+50)({\left(\frac{{x}_{i}}{i}\right)}^{k}-1)\right]}^{2}$
${f(x)}_{13}$	Perm 2	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(1, 1/2,\dots , 1/n\right)$	${[-d, d]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{\left[\sum\limits_{j=1}^{d}({j}^{i}+10)({\left({x}_{j}^{i}-\frac{1}{{j}^{i}}\right)}^{i}\right]}^{2}$
${f(x)}_{14}$	Plateau	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=30; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-5.12, 5.12]}^{d}$	30	$f\left(x\right)=30+\sum\limits_{i=1}^{d}|{x}_{i}|$
${f(x)}_{15}$	Powell	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-4, 5]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^\frac{d}{4}\left[{({x}_{4i-3}+10{x}_{4i-2})}^{2}+5{({x}_{4i-1}-{x}_{4i})}^{2}+{({x}_{4i-2}-{x}_{4i-1})}^{4}+10{({x}_{4i-3}-{x}_{4i})}^{4}\right]$
${f(x)}_{16}$	Quing	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-1.28, 1.28]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{({x}_{i}^{2}-i)}^{2}$
${f(x)}_{17}$	Quartic	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(-1,\dots , -1\right)$	${[-10, 10]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}i{x}_{i}^{4}+rand[\mathrm{0,1})$
${f(x)}_{18}$	Quintic	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-5.12, 5.12]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}|{x}_{i}^{5}-3{x}_{i}^{4}+4{x}_{i}^{3}+2{x}_{i}^{2}-10{x}_{i}-4|$
${f(x)}_{19}$	Rastrigin	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(1,\dots , 1\right)$	${[-5, 10]}^{d}$	30	$f\left(x\right)=10d+\sum\limits_{i=1}^{d}[{x}_{i}^{2}-10{\text{cos}}(2\pi {x}_{i})]$
${f(x)}_{20}$	Rosenbrock	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0.5,\dots , 0.5\right)$	${[-100, 100]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}100{({x}_{i+1}-{{x}_{i}}^{2})}^{2}+{({x}_{i}-1)}^{2}$
${f(x)}_{21}$	Schwefel 21	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-100, 100]}^{d}$	30	$f\left(x\right)=max\left\{\left|{x}_{i}\right|, 1\le i\le d\right\}$
${f(x)}_{22}$	Schwefel 22	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-100, 100]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}\left|{x}_{i}\right|+\prod_{i=1}^{d}\left|{x}_{i}\right|$
${f(x)}_{23}$	Step	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-100, 100]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}|{x}_{i}^{2}|$
${f(x)}_{24}$	Stybtang	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=-39.1659n; {\mathbf{x}}^{\mathbf{*}}=\left(-2.90,\dots , 2.90\right)$	${[-5, 5]}^{d}$	30	$f\left(x\right)=\frac{1}{2}\sum\limits_{i=1}^{d}({x}_{i}^{4}-16{x}_{i}^{2}+5{x}_{i})$
${f(x)}_{25}$	Trid	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=-n(n+4)(n-1)/6; {\mathbf{x}}^{\mathbf{*}}=\left[i(n+1-i)\right]\mathrm{ for} i=1,\dots ,n$	${[{-d}^{2}, {d}^{2}]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{({x}_{i}-1)}^{2}-\sum\limits_{i=2}^{d}{x}_{i}{x}_{i-1}$
${f(x)}_{26}$	Vincent	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=-n; {\mathbf{x}}^{\mathbf{*}}=\left(7.70,\dots , 7.70\right)$	${[0.25, 10]}^{d}$	30	$f\left(x\right)=-\frac{1}{n}\sum\limits_{i=1}^{n}{\text{sin}}[10{\text{log}}({x}_{i})]$
${f(x)}_{27}$	Zakharov	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-5, 10]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{2}+{\left(\sum\limits_{i=1}^{d}0.5i{x}_{i}\right)}^{2}+{\left(\sum\limits_{i=1}^{d}0.5i{x}_{i}\right)}^{4}$
${f(x)}_{28}$	Sphere	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=0,\dots , 0$	${[-5, 5]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{2}$
${f(x)}_{29}$	Sumpow	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-1, 1]}^{d}$	30	$f\left(x\right)=\sum\limits_{i=1}^{d}{|{x}_{i}|}^{i+1}$
${f(x)}_{30}$	Rastringin + Schwefel22 + Sphere	$f\left({\mathbf{x}}^{\mathbf{*}}\right)=0; {\mathbf{x}}^{\mathbf{*}}=\left(0,\dots , 0\right)$	${[-100, 100]}^{d}$	30	$f\left(x\right)=\left[10d+\sum\limits_{i=1}^{d}[{x}_{i}^{2}-10{\text{cos}}(2\pi {x}_{i})]\right]+\left[\sum\limits_{i=1}^{d}\left|{x}_{i}\right|+\prod_{i=1}^{d}\left|{x}_{i}\right|\right]+\left[\sum\limits_{i=1}^{d}{x}_{i}^{2}\right]$