Quantum-enhanced multiobjective large-scale optimization via parallelism

Highlights

• The quantum-based position update strategy in QPSO is integrated into the DE operator of DPCCMOLSEA framework.

• The novel variants PMLEA-QjDE and PMLEA-QJADE are presented, in which the quantum parameters are adaptive.

• The integration of parallel operation based on MPI substantially reduces the runtime of the quantum-enhanced algorithm.

Abstract

Traditional quantum-based evolutionary algorithms are intended to solve single-objective optimization problems or multiobjective small-scale optimization problems. However, multiobjective large-scale optimization problems are continuously emerging in the big-data era. Therefore, the research in this paper, which focuses on combining quantum mechanics with multiobjective large-scale optimization algorithms, will be beneficial to the study of quantum-based evolutionary algorithms. In traditional quantum-behaved particle swarm optimization (QPSO), particle position uncertainty prevents the algorithm from easily falling into local optima. Inspired by the uncertainty principle of position, the authors propose quantum-enhanced multiobjective large-scale algorithms, which are parallel multiobjective large-scale evolutionary algorithms (PMLEAs). Specifically, PMLEA-QDE, PMLEA-QjDE and PMLEA-QJADE are proposed by introducing the search mechanism of the individual particle from QPSO into differential evolution (DE), differential evolution with self-adapting control parameters (jDE) and adaptive differential evolution with optional external archive (JADE). Moreover, the proposed algorithms are implemented with parallelism to improve the optimization efficiency. Verifications performed on several test suites indicate that the proposed quantum-enhanced algorithms are superior to the state-of-the-art algorithms in terms of both effectiveness and efficiency.

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:

• 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.

• Based on jDE and JADE, we propose the variants PMLEA-QjDE and PMLEA-QJADE, in which the adaptive parameters are quantized.

• 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.