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International Conference on Mechatronics and Automation

Real-coded Quantum Evolutionary Algorithm for Complex Functions with High-dimension

Zhang Rui, Hui Gao

Published Aug 1, 2007• 15 citations
Publisher

Abstract

For optimzing complex functions with high-dimension, a real-coded quantum evolutionary algorithm (RCQEA) is proposed on the basis of the concept and principles of quantum computing such as qubits and superposition of states. Firstly, in this algorithm, real-coded triploid chromosomes, whose alleles are composed of real variable and a pair of probability amplitudes of the correspinding states of one qubit, are constructed to keep the diversity of solution. Secondly, complementary double mutation operator (CDMO), which is designed according to a pair of probability amplitudes of the correspinding states of one qubit satisfying the normalization condition, as well as quantum rotation gate (QRG) are used to update chromosomes, which can treat the balance between exploration and exploitation. Thirdly, discrete crossover (DC) is employed to expand search space. Finally, "Hill-climbing" selection (HCS) is adopted to accelerate the convergence speed. Simulation results on 4 benchmark complex functions with high-dimension show that RCQEA is not only effective, efficient, but also very adaptive to the dimensions, and has the characteristics of rapider convergence, more powerful global search capability and better stability.

DOI
10.1109/icma.2007.4304033

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For optimzing complex functions with high-dimension, a real-coded quantum evolutionary algorithm (RCQEA) is proposed on the basis of the concept and principles of quantum computing such as qubits and superposition of states. Firstly, in this algorithm, real-coded triploid chromosomes, whose alleles are composed of real variable and a pair of probability amplitudes of the correspinding states of one qubit, are constructed to keep the diversity of solution. Secondly, complementary double mutation operator (CDMO), which is designed according to a pair of probability amplitudes of the correspinding states of one qubit satisfying the normalization condition, as well as quantum rotation gate (QRG) are used to update chromosomes, which can treat the balance between exploration and exploitation. Thirdly, discrete crossover (DC) is employed to expand search space. Finally, "Hill-climbing" selection (HCS) is adopted to accelerate the convergence speed. Simulation results on 4 benchmark complex functions with high-dimension show that RCQEA is not only effective, efficient, but also very adaptive to the dimensions, and has the characteristics of rapider convergence, more powerful global search capability and better stability.
In order to optimize the complex functions,a real-coded quantum evolutionary algorithm is proposed based on the relational concepts and principles of quantum computing.Real-coded triploid chromosomes,whose alleles are composed of a component of the independent variable vector and a pair of probability amplitudes of the corresponding states of a qubit,are constructed to keep the population diversity.The complementary double mutation operator,which is designed according to the probability amplitudes of a qubit fulfilling the normalization conditions,and the quantum rotation gate are used to update chromosomes and realize a good balance between exploration and exploitation.Simulation results on benchmark functions show that the algorithm is well suitable for the complex function optimization,and has the characteristics of rapider convergence,more powerful global search capability and better stability.
In order to optimize the numerical functions with the continuous solution space, a new phase angle encoded quantum evolutionary algorithm(PAQEA) is proposed based on the quantum computing and real encoded evolutionary algorithm. In PAQEA, a quantum chromosome with probabilistic representation can represent all the feasible solution probabilistically, and the dynamic quantum rotation gate is used to update chromosomes. Thus, the population diversity and directional evolution realize a good balance between exploration and exploitation. Theoretical analysis shows that the PAQEA is a global convergence algorithm. Simulation results show that the algorithm is suitable for the optimization of complex numerical functions, and has the characteristics of rapider convergence, powerful global search capability and better stability.
Quantum evolutionary algorithm is a novel and efficient algorithm which inspired by the ideas of quantum computing,and it has excellent results on the computational efficiency and avoid falling into local extremum problems.Therefore,the combination of quantum mechanics and intelligent optimization algorithms will further expand the application areas of intelligent optimization algorithms to improve the optimization algorithm problem-solving ability.For this,the introduction of quantum computation to the differential evolution algorithm,we proposed a new evolutionary algorithm-quantum differential evolution algorithm.This method is the probability of quantum bits used in the real-coded chromosomes,with the quantum mutation,quantum crossover and quantum choose operation to achieve the update of chromosomal location,and using the quantum NOT gate to swap two probability amplitude of quantum bits,it can prevent the premature algorithm,while the algorithm converges faster.The function extreme value and TSP problems as an example to verify the effectiveness of the proposed algorithm.
For improving convergence rate and preventing prematurity in quantum evolutionary algorithm, an allele real-coded quantum evolutionary algorithm based on hybrid updating strategy is presented. The real variables are coded with probability superposition of allele. A hybrid updating strategy balancing the global search and local search is presented in which the superior allele is defined. On the basis of superior allele and inferior allele, a guided evolutionary process as well as updating allele with variable scale contraction is adopted. And H ε gate is introduced to prevent prematurity. Furthermore, the global convergence of proposed algorithm is proved by Markov chain. Finally, the proposed algorithm is compared with genetic algorithm, quantum evolutionary algorithm, and double chains quantum genetic algorithm in solving continuous optimization problem, and the experimental results verify the advantages on convergence rate and search accuracy.
Optimization method is important in engineering design and application. Quantum genetic algorithm has the characteristics of good population diversity, rapid convergence and good global search capability and so on. It combines quantum algorithm with genetic algorithm. A novel quantum genetic algorithm is proposed, which is called Variable-boundary-coded Quantum Genetic Algorithm (vbQGA) in which qubit chromosomes are collapsed into variable-boundary-coded chromosomes instead of binary-coded chromosomes. Therefore much shorter chromosome strings can be gained. The method of encoding and decoding of chromosome is first described before a new adaptive selection scheme for angle parameters used for rotation gate is put forward based on the core ideas and principles of quantum computation. Eight typical functions are selected to optimize to evaluate the effectiveness and performance of vbQGA against standard Genetic Algorithm (sGA) and Genetic Quantum Algorithm (GQA). The simulation results show that vbQGA is significantly superior to sGA in all aspects and outperforms GQA in robustness and solving velocity, especially for multidimensional and complicated functions.
Quantum genetic algorithms(QGA) typically construct traditional quantum chromosomes as independent quantum bit systems, which has facilitated the enhancement of search speed and precision in genetic algorithms, demonstrating powerful problem-solving capabilities in optimization issues. Although traditional QGA have enriched population diversity, they still tend to fall into local optima, and the evolutionary update operations require rotation directions to be determined by lookup tables, a process involving multiple conditional judgments that affect algorithm efficiency. To address these issues, an improved method has been proposed to further enhance the performance of QGA. In the initial population selection, a population grouping evolution method based on microhabitat coevolution strategy is used to prevent the algorithm from getting trapped in local optima when solving complex function problems. Additionally, quantum entanglement is introduced on top of quantum super-position by implementing quantum entanglement using quantum registers, and higher-order quantum rotation gates are employed that do not involve multiple conditional judgments of lookup tables, thus improving the efficiency of quantum chromosome measurement and evolutionary updates. Finally, traditional QGA and higher-order quantum genetic algorithms were applied to test functions in multiple experiments. The results indicate that, compared to traditional QGA, the higher-order quantum genetic algorithms have achieved an increase in convergence speed, an approximate 14.3% improvement in convergence precision, and possess enhanced global search capabilities.