A Modified QuantumInspired Genetic Algorithm Using Lengthening Chromosome Size and an Adaptive LookUp Table to Avoid Local Optima
Abstract
:1. Introduction
 Lengthening Chromosomes Size: DQGA increases the size of chromosomes throughout the algorithm run. This strategy leads to increasing precision levels for the duration of generations. Low precision levels for early generations cause higher global focus and less attention to detail, favoring diversification. As opposed to that, higher precision in the last generations promotes intensification. This manner guarantees a smooth shift from the exploration phase to the exploitation phase. It should be noted that the concept of utilizing variable chromosome size was introduced in [48] as an attempt to find a suitable chromosome size for reducing computational time. Also, in [49], the authors used different chromosome sizes to cover diverse coarsegrained and finegrained parts of a design in topological order. However, in this paper, we utilized incrementing chromosome size for different purposes, namely local optima and premature convergence avoidance.
 Adaptive Rotation Steps: Unlike the lookup table of the original GQA, which consists of fixed values for all generations and ignores the current state of the qubits, the proposed DQGA uses an adaptive lookup table which helps the algorithm to search more properly and improves the exploration–exploitation transition.
2. Fundamentals
2.1. Quantum Computing Basics
2.2. GQA
3. DQGA
3.1. Lengthening Chromosome Size Strategy
Algorithm 1 The pseudocode of DQGA 

${\mathit{x}}_{\mathit{i}}$  ${\mathit{b}}_{\mathit{i}}$  $\mathit{f}\left(\mathit{x}\right)\ge \mathit{f}\left(\mathit{b}\right)$  $\mathit{\Delta}{\mathit{\theta}}_{\mathit{i}}$ 

0  0  false  Equation (15) 
0  0  true  Equation (15) 
0  1  false  Equation (10) 
0  1  true  Equation (12) 
1  0  false  Equation (11) 
1  0  true  Equation (13) 
1  1  false  Equation (14) 
1  1  true  Equation (14) 
3.2. LookUp Table with Adaptive Rotation Steps
 When the $ith$ bit of the best fitted binary solution of the previous generation $\mathbf{b}$ and current chromosome $\mathbf{x}$ are not equal, and $\mathbf{b}$ is more fitted than $\mathbf{x}$, we rotate the corresponding qubit state in a direction that makes it more likely to collapse into the state of ${b}_{i}$ with a huge step. The rotation size of a huge step is formulated in Equation (10) for ${b}_{i}=1$ and ${x}_{i}=0$ and in and Equation (11) for ${b}_{i}=0$ and ${x}_{i}=1$.$$\Delta {\theta}_{i}=(\pi /2{\theta}_{i})\times m\phantom{\rule{12.0pt}{0ex}}$$$$\Delta {\theta}_{i}={\theta}_{i}\times m,$$
 When the $ith$ bit of the best fitted individual b and current chromosome $\mathbf{x}$ are different and x has a higher fitness value in comparison to b, the corresponding qubit is pushed to the state of ${x}_{i}$ but this time with a little caution or hesitation, as the previous iteration’s best individual guides us conversely. This leads to a relatively smaller rotation size, called medium step. Equations (12) and (13) show the mathematical representation of the case with ${b}_{i}=1$ and ${x}_{i}=0$ and the case with ${b}_{i}=0$ and ${x}_{i}=1$, respectively.$$\Delta {\theta}_{i}=(\pi /2{\theta}_{i})\times \frac{m}{20}$$$$\Delta {\theta}_{i}={\theta}_{i}\times \frac{m}{20}.$$
 The last case is when ${b}_{i}$ and ${x}_{i}$ are identical. In this case, we do not care about which individual yields better fitness, as both of them share a similar state. So, we just move the qubit state by a tiny step in order to slightly confirm the last iteration’s best individual state regardless of the fitness comparison. These minor fluctuations help to keep the diversity of the population. Equation (14) expresses the tiny step when ${b}_{i}$ and ${x}_{i}$ are in state ‘1’, while Equation 15 shows otherwise.$$\Delta {\theta}_{i}=(\pi /2{\theta}_{i})\times \frac{m}{500}$$$$\Delta {\theta}_{i}={\theta}_{i}\times \frac{m}{500}.$$
3.3. Distribution of Generations in Different Precision Levels
4. Experimental Results and Comparison Discussion
4.1. Testing DQGA on Benchmark Functions
4.2. Constrained Engineering Design Optimization Using DQGA
4.2.1. Pressure Vessel Design
4.2.2. Speed Reducer Design
4.2.3. Cantilever Beam Design
5. Conclusions and Potential Future Work
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
References
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Function Name  Function Description  Domain  ${\mathit{F}}_{\mathbf{min}}$ 

Sphere Function  ${F}_{1}\left(\mathbf{x}\right)={\sum}_{i=1}^{n}{x}_{i}^{2}$  $[5.12,5.12]$  0 
Schwefel 2.22  ${F}_{2}\left(\mathbf{x}\right)={\sum}_{i=1}^{n}\left{x}_{i}\right+{\prod}_{i=1}^{n}\left{x}_{i}\right$  $[10,10]$  0 
Schwefel 2.21  ${F}_{3}\left(\mathbf{x}\right)={max}_{i}\left\{\left{x}_{i}\right,1\le i\le n\right\}$  $[1.28,1.28]$  0 
Rosenbrock  ${F}_{4}\left(\mathbf{x}\right)={\sum}_{i=1}^{n1}\left[100{\left({x}_{i+1}{x}_{i}^{2}\right)}^{2}+{({x}_{i}1)}^{2}\right]$  $[30,30]$  0 
Step Function  ${F}_{5}\left(\mathbf{x}\right)={\sum}_{i=1}^{n}{\left(\u230a{x}_{i}+0.5\u230b\right)}^{2}$  $[100,100]$  0 
Ackley  ${F}_{6}\left(\mathbf{x}\right)=20exp(0.2\sqrt{\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}^{2}})$ $\phantom{\rule{1.em}{0ex}}exp(\frac{1}{n}{\sum}_{i=1}^{n}cos2\pi {x}_{i})+20+e$  $[32,32]$  0 
Rastrigin  ${F}_{7}\left(\mathbf{x}\right)=10n+{\sum}_{i=1}^{n}\left[{x}_{i}^{2}10cos\left(2\pi {x}_{i}\right)\right]$  $[5.12,5.12]$  0 
Schwefel  ${F}_{8}\left(\mathbf{x}\right)=418.9829n{\sum}_{i=1}^{n}{x}_{i}sin\sqrt{\left{x}_{i}\right}$  $[500,500]$  0 
Styblisky–Tang  ${F}_{9}\left(\mathbf{x}\right)=\frac{1}{2}{\sum}_{i=1}^{n}({x}_{i}^{4}16{x}_{i}^{2}+5{x}_{i})$  $[5,5]$  $39.16599n$ 
Levy  ${F}_{10}\left(\mathbf{x}\right)={sin}^{2}\left(\pi {\omega}_{1}\right)+{\sum}_{i=1}^{n1}{({\omega}_{i}1)}^{2}\times $ $\phantom{\rule{1.em}{0ex}}\left[1+10{sin}^{2}(\pi {\omega}_{i}+1)\right]+$ $\phantom{\rule{1.em}{0ex}}{({\omega}_{n}1)}^{2}\left[1+{sin}^{2}\left(2\pi {\omega}_{n}\right)\right],$ $\phantom{\rule{1.em}{0ex}}where\phantom{\rule{1.em}{0ex}}{\omega}_{i}=1+\frac{{x}_{i}1}{4},for\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,\dots ,n$  $[10,10]$  0 
Algorithm  Parameter  Value 

GA [54]  Implementation type  Realcoded 
Selection method  Roulette wheel  
Crossover probability  $80\%$  
Mutation method  Flip  
Mutation probability  $0.05\%$  
GQA [8]  No Parameter setting  
PSO [55]  ${c}_{1}$  2 
${c}_{2}$  2  
weight_min  0.1  
weight_max  0.9  
QPSO [56]  ${a}_{1}$  1 
${a}_{2}$  0.5  
MFO [57]  a  $[2,1]$ 
b  1  
DQGA  min_length  16 
max_length  32  
interval  4  
a  1.1  
b  0.1 
F  GA [54]  GQA [8]  PSO [55]  QPSO [56]  MFO [57]  DQGA  

${F}_{1}$  Mean STD  $2.90\times {10}^{0}$ $4.06\times {10}^{1}$  $2.32\times {10}^{1}$ $9.40\times {10}^{0}$  $5.58\times {10}^{1}$ $2.76\times {10}^{1}$  $2.39\times {10}^{2}$ $3.38\times {10}^{2}$  $\mathbf{1}.\mathbf{28}\times {\mathbf{10}}^{\mathbf{5}}$ $2.53\times {10}^{5}$  $3.05\times {10}^{4}$ $2.02\times {10}^{4}$ 
${F}_{2}$  Mean STD  $1.17\times {10}^{1}$ $1.03\times {10}^{0}$  $3.34\times {10}^{1}$ $1.01\times {10}^{1}$  $2.48\times {10}^{5}$ $7.75\times {10}^{5}$  $3.95\times {10}^{0}$ $5.46\times {10}^{0}$  $4.35\times {10}^{0}$ $9.71\times {10}^{0}$  $\mathbf{7}.\mathbf{41}\times {\mathbf{10}}^{\mathbf{2}}$ $2.44\times {10}^{2}$ 
${F}_{3}$  Mean STD  $1.24\times {10}^{1}$ $1.30\times {10}^{0}$  $5.70\times {10}^{1}$ $6.53\times {10}^{0}$  $3.65\times {10}^{1}$ $5.26\times {10}^{0}$  $1.67\times {10}^{1}$ $5.07\times {10}^{0}$  $4.94\times {10}^{1}$ $1.41\times {10}^{1}$  $\mathbf{7}.\mathbf{86}\times {\mathbf{10}}^{\mathbf{0}}$ $2.43\times {10}^{0}$ 
${F}_{4}$  Mean STD  $1.93\times {10}^{5}$ $6.31\times {10}^{4}$  $8.17\times {10}^{6}$ $5.17\times {10}^{6}$  $4.04\times {10}^{7}$ $4.50\times {10}^{7}$  $7.93\times {10}^{2}$ $8.27\times {10}^{2}$  $9.76\times {10}^{3}$ $2.72\times {10}^{4}$  $\mathbf{5}.\mathbf{56}\times {\mathbf{10}}^{\mathbf{2}}$ $7.50\times {10}^{2}$ 
${F}_{5}$  Mean STD  $1.17\times {10}^{3}$ $1.72\times {10}^{2}$  $8.70\times {10}^{3}$ $4.40\times {10}^{3}$  $2.31\times {10}^{4}$ $1.07\times {10}^{4}$  $5.59\times {10}^{1}$ $7.11\times {10}^{1}$  $7.63\times {10}^{0}$ $5.01\times {10}^{0}$  $\mathbf{2}.\mathbf{37}\times {\mathbf{10}}^{\mathbf{0}}$ $1.71\times {10}^{0}$ 
${F}_{6}$  Mean STD  $7.32\times {10}^{0}$ $4.11\times {10}^{1}$  $1.49\times {10}^{1}$ $1.43\times {10}^{0}$  $1.99\times {10}^{1}$ $1.49\times {10}^{1}$  $3.92\times {10}^{0}$ $1.55\times {10}^{0}$  $1.76\times {10}^{1}$ $5.06\times {10}^{0}$  $\mathbf{1}.\mathbf{63}\times {\mathbf{10}}^{\mathbf{0}}$ $7.56\times {10}^{1}$ 
${F}_{7}$  Mean STD  $1.12\times {10}^{2}$ $9.79\times {10}^{0}$  $1.64\times {10}^{2}$ $2.89\times {10}^{1}$  $2.82\times {10}^{2}$ $6.10\times {10}^{1}$  $7.55\times {10}^{1}$ $2.49\times {10}^{1}$  1.31E+02 $2.82\times {10}^{1}$  $\mathbf{4}.\mathbf{94}\times {\mathbf{10}}^{\mathbf{1}}$ $9.28\times {10}^{0}$ 
${F}_{8}$  Mean STD  $5.67\times {10}^{3}$ $3.58\times {10}^{2}$  $4.63\times {10}^{3}$ $5.80\times {10}^{2}$  $7.29\times {10}^{3}$ $1.15\times {10}^{3}$  $3.28\times {10}^{3}$ $4.28\times {10}^{2}$  $4.29\times {10}^{3}$ $4.63\times {10}^{2}$  $\mathbf{1}.\mathbf{91}\times {\mathbf{10}}^{\mathbf{3}}$ $4.03\times {10}^{2}$ 
${F}_{9}$  Mean STD  $9.79\times {10}^{2}$ $2.30\times {10}^{1}$  $9.66\times {10}^{2}$ $4.96\times {10}^{1}$  $7.77\times {10}^{2}$ $5.26\times {10}^{1}$  $1.04\times {10}^{3}$ $3.77\times {10}^{1}$  $1.03\times {10}^{3}$ $4.01\times {10}^{1}$  $\mathbf{1}.\mathbf{14}\times {\mathbf{10}}^{\mathbf{3}}$ $5.66\times {10}^{0}$ 
${F}_{10}$  Mean STD  $3.54\times {10}^{0}$ $6.90\times {10}^{1}$  $2.88\times {10}^{1}$ $1.04\times {10}^{1}$  $8.60\times {10}^{1}$ $3.30\times {10}^{1}$  $5.45\times {10}^{0}$ $3.18\times {10}^{0}$  $8.45\times {10}^{0}$ $4.75\times {10}^{0}$  $\mathbf{3}.\mathbf{27}\times {\mathbf{10}}^{\mathbf{0}}$ $1.85\times {10}^{0}$ 
Function  GA [54]  GQA [8]  PSO [55]  QPSO [56]  MFO [57] 

${F}_{1}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$ 
${F}_{2}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $3.82\times {10}^{1}$ 
${F}_{3}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$ 
${F}_{4}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $6.14\times {10}^{1}$  $3.29\times {10}^{1}$ 
${F}_{5}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $3.42\times {10}^{6}$ 
${F}_{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $2.60\times {10}^{6}$  $1.73\times {10}^{6}$ 
${F}_{7}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.15\times {10}^{4}$  $1.73\times {10}^{6}$ 
${F}_{8}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.92\times {10}^{6}$  $1.73\times {10}^{6}$ 
${F}_{9}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$ 
${F}_{10}$  $7.73\times {10}^{3}$  $1.73\times {10}^{6}$  $1.73\times {10}^{6}$  $6.16\times {10}^{4}$  $2.16\times {10}^{5}$ 
Algorithm  ${\mathit{T}}_{\mathit{s}}$  ${\mathit{T}}_{\mathit{h}}$  R  L  Minimum Weight 

Branchbound [60]  1.125  0.625  48.97  106.72  7982.5 
GA [66]  0.81250  0.43750  42.097398  176.65405  6059.94634 
GWO [61]  0.812500  0.434500  42.089181  176.758731  6051.5639 
WOA [62]  0.812500  0.437500  42.0982699  176.638998  6059.7410 
HHO [63]  0.81758383  0.4072927  42.09174576  176.7196352  6000.46259 
WSA [64]  0.78654289  0.39348835  40.75268075  194.78059812  5929.62188231 
AOA [65]  0.8303737  0.4162057  42.75127  169.3454  6048.7844 
DQGA  0.79760749  0.39427185  41.31109227  186.64366007  5921.48841641 
Algorithm  ${\mathit{x}}_{2}$  ${\mathit{x}}_{3}$  ${\mathit{x}}_{4}$  ${\mathit{x}}_{5}$  ${\mathit{x}}_{6}$  ${\mathit{x}}_{7}$  ${\mathit{x}}_{8}$  Minimum Weight 

CS [68]  3.5015  0.7000  17  7.6050  7.8181  3.3520  5.2875  3000.9810 
FA [69]  3.507495  0.7001  17  7.7196  8.0808  3.351512  5.287051  3010.137492 
WSA [64]  3.500000  0.7  17  7.3  7.8  3.350215  5.286683  2996.348222 
hHHOSCA [70]  3.506119  0.7  17  7.3  7.9914  3.452569  5.286749  3029.873076 
AAO [71]  3.4999  0.6999  17  7.3  7.8  3.3502  5.2872  2996.783 
AO [72]  3.5021  0.7000  17  7.3099  7.7476  3.3641  5.2994  3007.7328 
AOA [65]  3.50384  0.7  17  7.3  7.7293  3.35649  5.2867  2997.9157 
DQGA  3.500024  0.7  17  7.3  7.8  3.350226  5.286621  2996.321084 
Algorithm  ${\mathit{x}}_{1}$  ${\mathit{x}}_{2}$  ${\mathit{x}}_{3}$  ${\mathit{x}}_{4}$  ${\mathit{x}}_{5}$  Minimum Weight 

CS [68]  6.0089  5.3049  4.5023  3.5077  2.1504  1.33999 
SOS [73]  6.01878  5.30344  4.49587  3.49896  2.15564  1.33996 
MFO [57]  5.984872  5.316727  4.497333  3.513616  2.161620  1.339988 
GCA_I [74]  6.01  5.304  4.49  3.498  2.15  1.34 
GCA_II [74]  6.01  5.304  4.49  3.498  2.15  1.34 
SMA [75]  6.017757  5.310892  4.493758  3.501106  2.150159  1.33996 
AO [72]  5.8881  5.5451  4.3798  3.5973  2.1026  1.3390 
DQGA  5.967485  4.821212  4.502603  3.488657  2.161575  1.306752 
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Hakemi, S.; Houshmand, M.; Hosseini, S.A.; Zhou, X. A Modified QuantumInspired Genetic Algorithm Using Lengthening Chromosome Size and an Adaptive LookUp Table to Avoid Local Optima. Axioms 2023, 12, 978. https://doi.org/10.3390/axioms12100978
Hakemi S, Houshmand M, Hosseini SA, Zhou X. A Modified QuantumInspired Genetic Algorithm Using Lengthening Chromosome Size and an Adaptive LookUp Table to Avoid Local Optima. Axioms. 2023; 12(10):978. https://doi.org/10.3390/axioms12100978
Chicago/Turabian StyleHakemi, Shahin, Mahboobeh Houshmand, Seyyed Abed Hosseini, and Xujuan Zhou. 2023. "A Modified QuantumInspired Genetic Algorithm Using Lengthening Chromosome Size and an Adaptive LookUp Table to Avoid Local Optima" Axioms 12, no. 10: 978. https://doi.org/10.3390/axioms12100978