Quantum-enhanced multiobjective large-scale optimization via parallelism
Introduction
The quantum-inspired evolutionary algorithm (QIEA) combines the evolutionary algorithm (EA) and quantum computation, achieving a balance between exploration and exploitation [1]. Compared with EAs, QIEAs use the probability amplitude representation of qubits to encode chromosomes. The use of the quantum rotation gate update strategy allows QIEAs to converge more quickly [2]. Quantum gate updating is a key step in quantum evolutionary algorithms (QEAs). Xiong et al. [3] summarized the most commonly used quantum rotation gates. The superposition and entanglement of the quantum state provides QIEAs with the potential to apply parallelism in the process of evolution [4]. Patvardhan et al. [4] proposed a parallel improved quantum inspired evolutionary algorithm (IQIEA-P) with a high acceleration ratio for large-size quadratic knapsack problems, which have only one objective.
In addition to single-objective optimization problems, many real-world problems need to optimize multiple conflicting objectives simultaneously. Problems with two or three objectives are usually called multiobjective optimization problems (MOPs). Problems with more than three objectives are called many-objective optimization problems (MaOPs). Moreover, many practical optimization problems have hundreds of decision variables [5,6], which are referred to as large-scale optimization problems. Problems with two or three objectives and a large number of decision variables (usually more than 100) are denoted as multiobjective large-scale optimization problems (MOLSOPs).
Considering the excellent diversity characteristics of quantum systems, many studies have combined quantum computation with single-objective EAs and applied them to numerical optimization [7], combinatorial optimization [8], production scheduling [9], vehicle routing [10], and other fields. Pavithr [11] proposed a hybrid quantum-inspired social evolutionary algorithm (QSE) that performed well on the 0–1 knapsack problem. Dahi et al. [12] proposed a quantum-inspired genetic algorithm (QIGA) with new quantum gates to address the antenna positioning problem. Alanis et al. [13] proposed a nondominated quantum optimization algorithm (NDQO) to optimize a multiobjective routing problem. Li et al. [14] proposed a quantum memetic algorithm (QMA) by introducing cultural evolution. Some scholars have combined differential evolution (DE) [15,16] with quantum computation. Hu et al. [17] combined quantum-behaved particle swarm optimization (QPSO) [18], DE and the tabu search algorithm [19], proposing the hybridized vector optimal algorithm QPSO-DET, which better balances the relationship between local search and global search. SaiToh et al. [20] showed that even with the introduction of quantum mutation operators, the algorithm is sometimes prone to fall into local search. Therefore, a quantum crossover process that crosses all chromosomes in each generation was proposed. Based on QEAs, Ren et al. [21] proposed a hybrid quantum differential evolution algorithm (HQDE) that updates quantum chromosomes by quantum differential evolution (QDE) and quantum harmony search (QHS).
However, quantum theory has rarely been applied to solve large-scale optimization problems. Ding et al. [22] proposed a single-objective quantum cooperative coevolution algorithm for attribute reduction (QCCAR) with respect to large data sets by combining the cooperative coevolutionary (CC) [23] framework with a QEA. Tian et al. [24] combined the QPSO algorithm with the CC framework and proposed the single-objective QPSO_CC framework to solve large-scale optimization problems. They used the random decomposition strategy to separate the search space and used QPSO to optimize each subgroup. Fang et al. [25] proposed a random selection decomposition strategy based on random dimension reduction to solve large-scale optimization problems and proposed the RSQPSO algorithm based on the QPSO and random selection strategy. The above three algorithms have applied the CC framework and QIEA for large-scale optimization but only been used for single-objective large-scale optimization problems.
Traditional EAs have been applied in many fields [[26], [27], [28]], but their optimization performance substantially decreases as the number of decision variables increases. Research on multiobjective large-scale EAs is both popular and difficult [[29], [30], [31], [32]]. Among these algorithms, the variable grouping and CC strategy are helpful in improving the optimization performance with respect to large-scale problems.
Some scholars have combined quantum mechanics with multiobjective EAs. Kumari et al. proposed a quantum heuristic multiobjective differential evolution algorithm (QMDEA) [33] and a multiobjective quantum heuristic hybrid differential evolution algorithm (MQHDE) [34] to balance exploration and exploitation. All these methods have combined DE with a genetic algorithm (GA) and quantum computation to form multiobjective frameworks, contributing to the balance between convergence and diversity in multiobjective optimization algorithms [35]. Li et al. [36] proposed the quantum behavioral discrete multiobjective particle swarm optimization (QDM-PSO) algorithm and applied it to a large-scale complex network clustering problem. Mouradian et al. [37] modeled task allocation for a large number of robots in a large-scale natural environment as a multiobjective problem and proposed the quantum multiobjective particle swarm optimization (QMOPSO) algorithm. Mousavi et al. [38] used a QEA to solve the computational complexity of coalition formation in large-scale unmanned aerial vehicle (UAV) networks. Tang et al. [39] proposed a QPSO with memetic algorithm and memory (SMQPSO) algorithm to solve continuous nonlinear large-scale problems.
Distributed and parallel algorithms [40] can capitalize on large numbers of computing resources and substantially reduce algorithm time consumption, improving algorithm efficiency [41]. Tan et al. [42] proposed a distributed cooperative coevolutionary multiobjective optimization algorithm. Cao et al. proposed a distributed parallel cooperative coevolutionary multiobjective evolutionary algorithm (DPCCMOEA) [43] based on an improved variable analysis strategy and a distributed parallel cooperative coevolutionary multiobjective large-scale evolutionary algorithm (DPCCMOLSEA) [44] to solve MOLSOPs. Both algorithms are based on a decomposition strategy in which the variables are broken down into groups, and each group is optimized by one subpopulation using the DE operator [15,16]. Based on DPCCMOLSEA, we propose the parallel multiobjective large-scale evolutionary algorithm (PMLEA) with quantum-enhanced DE, quantum-enhanced differential evolution with self-adapting control parameters (jDE) and quantum-enhanced adaptive differential evolution with optional external archive (JADE), denoted as PMLEA-QDE, PMLEA-QjDE and PMLEA-QJADE, respectively.
The contributions of the present study include the following:
- 1.
We integrate the position update strategy based on the theory of quantum mechanics in QPSO into the DE operator of the DPCCMOLSEA framework to optimize the population.
- 2.
Based on jDE and JADE, we propose the variants PMLEA-QjDE and PMLEA-QJADE, in which the adaptive parameters are quantized.
- 3.
The integration of parallel operation based on the message passing interface (MPI) substantially reduces the runtime of the quantum-enhanced algorithm.
The organization of this paper is as follows. Section 2 introduces the large-scale MOPs and the QPSO algorithm. The proposed methodology is described in Section 3. Section 4 reports the experimental comparison results and provides an analysis. Finally, Section 5 summarizes this paper.
Section snippets
MOLSOPs
MOPs in which the decision variable number is greater than or equal to 100 are called MOLSOPs. In general, an MOP with N decision variables and M objectives can be described as follows [34,45]:where x is a decision vector in decision space Ω, N ≥ 100, and F(x) is an objective vector located in the objective space, 1 < M ≤ 3.
Quantum-behaved particle swarm optimization
The QPSO algorithm is based on the quantum potential well model inspired by the principles of quantum mechanics. It
The proposed quantum-enhanced algorithm
In QPSO, the randomness of the particle position causes it to have better global search capability. Therefore, inspired by the theory of position update in QPSO and based on the DPCCMOLSEA framework, we propose PMLEA-QDE, PMLEA-QjDE and PMLEA-QJADE.
DPCCMOEA [43] and DPCCMOLSEA [44] both rely on decomposition to solve MOLSOPs. In this section, we describe DPCCMOEA, DPCCMOLSEA, and the proposed quantum-enhanced algorithms.
Experimental setup
We compared the proposed quantum-enhanced algorithms PMLEA-QDE, PMLEA-QjDE, and PMLEA-QJADE with PMLEA-DE, PMLEA-jDE, PMLEA-JADE, PMLEA-PSO, PMLEA-QPSO, the cooperative coevolutionary generalized differential evolution 3 (CCGDE3) algorithm [52], the multiobjective evolutionary algorithm based on decision variable analyses (MOEA/DVA) [53], MOEA/D [47], cooperative multiobjective differential evolution (CMODE) [54], nondominated sorting genetic algorithm II (NSGA-II) [55], weighted optimization
Conclusion
Based on the DPCCMOLSEA framework, we proposed a series of quantum-enhanced algorithms: PMLEA-QDE, PMLEA-QjDE and PMLEA-QJADE. We combined quantum theory and the DE operator to optimize the population. Moreover, in optimizers of jDE and JADE, the adaptive parameters are enhanced with quantum. We used the multiobjective test suites DTLZ, WFG, LSMOP and MaOP to compare the quantum-enhanced algorithms with the other state-of-the-art multiobjective algorithms and ranked the algorithms by using the
Author statement
We have made substantial contributions to the conception or design of the work. We have drafted the work or revised it critically for important intellectual content. We have approved the final version to be published. And we agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.
Declaration of competing interest
The authors declared that they have no conflicts of interest to this work.
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
We believe that this manuscript is appropriate for publication in the Swarm and Evolutionary Computation. This manuscript has not been published and is not under consideration for publication elsewhere. All authors have read and approved the final version of the manuscript.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 61976242, in part by the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University under Grant No. 2018002, in part by the Open Fund of Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, under Grant No. IPIU2019003, and in part by the State Key Program of National Natural Science
References (66)
- et al.
Quantum-inspired evolutionary algorithm for a class of combinatorial optimization
IEEE Trans. Evol. Comput.
(2002) Advances in quantum-inspired evolutionary algorithms
Control Decis.
(2008)- et al.
Quantum rotation gate in quantum-inspired evolutionary algorithm: a review, analysis and comparison study
Swarm Evol. Comput.
(2018) - et al.
Parallel improved quantum inspired evolutionary algorithm to solve large size quadratic knapsack problems
Swarm Evol. Comput.
(2016) - et al.
Applying graph-based differential grouping for multiobjective large-scale optimization
Swarm Evol. Comput.
(2020) - et al.
Semantic pooling for complex event analysis in untrimmed videos
IEEE Trans. Pattern Anal. Mach. Intell.
(2016) - et al.
Quantum-inspired space search algorithm (QSSA) for global numerical optimization
Appl. Math. Comput.
(2011) - et al.
Application of quantum-inspired binary gravitational search algorithm for thermal unit commitment with wind power integration
Energy Convers. Manag.
(2014) - et al.
Quantum-inspired evolutionary multiobjective optimization for a dynamic production scheduling approach
- et al.
A new improved quantum evolution algorithm with local search procedure for capacitated vehicle routing problem
Math. Probl Eng.
(2013)
Quantum inspired social evolution (QSE) algorithm for 0-1 knapsack problem
Swarm Evol. Comput.
A quantum-inspired genetic algorithm for solving the antenna positioning problem
Swarm Evol. Comput.
Quantum-assisted routing optimization for self-organizing networks
IEEE Access
Quantum memetic evolutionary algorithm-based low-complexity signal detection for underwater acoustic sensor networks
IEEE Trans. Syst. Man Cybern. Part C
Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces
J. Global Optim.
Differential Evolution: A Practical Approach to Global Optimization
A hybridized vector optimal algorithm for multi-objective optimal designs of electromagnetic devices
IEEE Trans. Magn.
Quantum-behaved particle swarm optimization: analysis of individual particle behavior and parameter selection
Evol. Comput.
A universal tabu search algorithm for global optimization of multimodal functions with continuous variables in electromagnetics
IEEE Trans. Magn.
A quantum genetic algorithm with quantum crossover and mutation operations
Quant. Inf. Process.
Hybrid quantum differential evolutionary algorithm and its applications
Control Theory & Appl.
A novel quantum cooperative co-evolutionary algorithm for large-scale minimum attribute reduction optimization
Cooperative coevolution: an architecture for evolving coadapted subcomponents
Evol. Comput.
Quantum-behaved particle swarm optimization with cooperative coevolution for large scale optimization
A novel quantum-behaved particle swarm optimization with random selection for large scale optimization
Software module clustering as a multi-objective search problem
IEEE Trans. Software Eng.
Rank-constrained spectral clustering with flexible embedding
IEEE Trans. Neural Netw. Learn. Syst.
A survey of multiobjective evolutionary algorithms for data mining: Part I
IEEE Trans. Evol. Comput.
Use of cooperative coevolution for solving large scale multiobjective optimization problems
A multiobjective evolutionary algorithm based on decision variable analyses for multiobjective optimization problems with large-scale variables
IEEE Trans. Evol. Comput.
A random-based dynamic grouping strategy for large scale multi-objective optimization
A decision variable clustering-based evolutionary algorithm for large-scale many-objective optimization
IEEE Trans. Evol. Comput.
Multi-objective test suite minimisation using quantum-inspired multi-objective differential evolution algorithm
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