Multi-scale Quantum Harmonic Oscillator Algorithm Research Articles

Bare-bones multi-scale quantum harmonic oscillator algorithm for global optimization

The Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA) is a novel heuristic optimization technique, underpinned by the quantum wave function as its theoretical foundation. In this study, we present a streamlined MQHOA framework, dubbed the Bare-Bones Multi-scale Quantum Harmonic Oscillator Algorithm (BBMQHOA). This minimalist paradigm employs the most rudimentary free particle in quantum mechanics to simulate particle movement. BBMQHOA leverages group stability over individual stability, enabling the algorithm to fully capitalize on population data, thereby enhancing exploration and exploitation. An extensive array of experiments is carried out on a collection of 24 benchmark functions and a pair of real-world applications to validate the efficacy of our suggested methodology. The outcomes reveal that its performance is both practical and competitive while simultaneously maintaining user-friendliness.

Quantum-inspired optimization algorithm with adaptive correction of energy position: Methodology and a case study

Efficient and stable global optimizers constitute a noteworthy arena of academic study and real-world applications. Since Multi-scale Quantum Harmonic Oscillator Algorithm inspired by the quantum motion for solving optimization problems was proposed, considerable contributions regarding this algorithm have been achieved in recent years. Nevertheless, issues such as the aggregation effect during sampling as well as recurrence and blindness in random searches hinder the performance of the algorithm. Motivated by this situation, a variant of Multi-scale Quantum Harmonic Oscillator Algorithm is put forward to improve the efficiency of the system convergence while maintaining the solution diversity. The measurement of the solution position through the collapse of the quantum state to the classical state is realized by means of quantum Monte Carlo simulations, and the energy position is established as a metric for energy observation. Then, the adaptive correction of the energy position is explored to improve algorithm performance. The core idea of our mechanism is to adaptively guide the candidate solutions toward convergence to the ground state by means of attractive factors based on the relationship among the energy positions of several reference points. Experimental results obtained on the CEC2013 benchmark functions and a real-world application indicate that the performance of our scheme is competitive and that it achieves prominence among the compared algorithms as the dimensionality increases.

Multiscale quantum harmonic oscillator algorithm with guiding information for single objective optimization

• This paper proposes a way to optimize MQHOA to further improve its performance. • The use of information in heuristic algorithms is discussed in this paper. It can be used to improve the performance of other heuristic optimization algorithms. • The proposed algorithm was evaluated on a large number of test functions. The multiscale quantum harmonic oscillator algorithm (MQHOA) is a competitive heuristic optimization algorithm that has been successfully implemented in many applications. This paper proposes a novel way to optimize MQHOA to further improve its performance. The idea is to use the historical information in the evolutionary iterative process of the algorithm to derive a direction in the quantum harmonic oscillator (QHO) process and as a multi scale in the M process, then take the guidance information as the parameter to generate a new solution. The combination of these processes, i.e., MQHOA combined with the guidance information, is called GI-MQHOA. The experimental results show that the guidance information is of great significance for the exploration and exploitation of MQHOA. The proposed algorithm was evaluated on the CEC2014 test suite, and shown to be comparable to other state-of-the-art swarm intelligence and heuristic algorithms. The principle of using guiding information is simple and effective and can be easily transplanted to other heuristic algorithms.

Multiscale quantum harmonic oscillator optimization algorithm with multiple quantum perturbations for numerical optimization

Multi-scale quantum harmonic oscillator algorithm (MQHOA) is a recent proposed intelligent algorithm, in which the optimization process can be regarded as the multi-scale quantum annealing process with respect to the constraint of the harmonic oscillator potential well. It has been proved effective and efficient to deal with unimodal and multimodal numerical optimization problems. However, it takes a long time for the particles system to reach the ground-state equilibrium at each annealing scale. Motivated by this situation, a diffusion Monte Carlo method based dynamic sampling regulation strategy is proposed to enhances the sampling efficiency by dynamically adjusting the sampling frequency. Moreover, the particles with different kinetic energies are introduced into the optimization system as quantum perturbations, which reduces the probability of the algorithm falling into a local optimum by keeping the different searching scales simultaneously. The theoretical derivations of this method are presented. The effectiveness of the algorithm is studied by comparing it with previous versions of the MQHOA and several famous intelligent algorithms on uni- and multimodal benchmark functions. The experimental results illustrate that the proposed method has a comparable performance for numerical optimization problems.

Exploring superposition state in multi-scale quantum harmonic oscillator algorithm

Research on the merging of quantum mechanics and intelligence algorithms has received extensive attention in recent years. When applying quantum mechanics to intelligent algorithms, the first noteworthy problem is the representation of solutions in the form of a probability superposition. In the recently proposed multi-scale quantum harmonic oscillator algorithm (MQHOA), the representation of the solutions is not a superposition of the probability distributions, but a single distribution. This incorrect representation of the superposition state reduces the diversity of the solutions and limits the performance of the algorithm. To alleviate this problem, in this paper, a multiple-population-based mechanism that uses a quantum superposition is presented for MQHOA, which increases the search capacity of the algorithm by enhancing the diversity of the solutions. With the proposed method, the eigenstate is represented by the different size and distribution subpopulations of the particles. The optimization process is the process of continually changing the size and the distribution of the subpopulations to make the superposition state collapse to the ground state. The multiple-population-based superposition state greatly increases the diversity of the optimization system and enhances the movement efficiency of the particles. The effectiveness of the method based on the performance of the algorithm was thoroughly investigated by comparing the method with the original MQHOA and several well-known intelligent algorithms on unimodal and multimodal benchmark functions and on a real-world charge management problem for plug-in hybrid electric vehicles. The experimental results reveal the effectiveness of the multiple-population-based superposition state used by the proposed method.

Relevance Feedback For Image Retrieval Using Transfer Learning and Improved MQHOA

Image retrieval is a challenging technology in multimedia applications where meeting the users’ subjective retrieval needs while achieving high retrieval performance is insufficient for existing methods. In this work, a related feedback image retrieval algorithm based on deep learning and optimization algorithm (CAMQHOA-RF) is proposed. Transfer learning based on the deep convolutional neural network is applied to extract deeper image features to reduce the semantic gap. The multi-scale quantum harmonic oscillator algorithm improved by the idea of “aggregation” is introduced to search the feature space effectively. The covariance matrix is used to strengthen the relationship between feature points at different scales to guide feature points to approach ideal query points faster. Moreover, the query point is reselected based on the feedback information to explore more potential users’ interest areas. Experiments have shown that compared with other algorithms, the proposed algorithm has fewer parameters that need to be set, but higher retrieval accuracy, faster retrieval speed, and stronger robustness are obtained, which can meet users better.

Impact of migration strategies and individual stabilization on multi-scale quantum harmonic oscillator algorithm for global numerical optimization problems

Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA) is a population-based metaheuristic algorithm proposed recently. It has been proved effective and efficient to deal with unimodal and multimodal problems. Although the mechanism of replacing the worst particle with the fittest individual in MQHOA helps to fasten the iteration process, it can easily lead to premature convergence. Instead of direct replacement, several migration strategies are proposed to maintain the diversity of the population and help to obtain the global optima in difficult function evaluations. The impacts of the migration strategies and individual stabilization on the improvement of the algorithms in their effectiveness, reliability, accuracy and efficiency are well researched. A variety of multi-dimensional unimodal and multimodal benchmark functions are applied to illustrate the optimization performance of the proposed algorithms. Some of the best competitors in MQHOAs with migration strategies are selected to compare with several state-of-the-art stochastic algorithms. Experimental results presented suggest some conclusions: First, the individual stabilization mechanism does not significantly improve the performance of MQHOA. Second, random migration does not obviously help MQHOA perform much better. Third, migration strategies significantly affect the performance of MQHOA, and some of MQHOAs with migration strategies are very competitive to deal with numerical optimization problems.

Unusual phenomenon of optimizing the Griewank function with the increase of dimension

The Griewank function is a typical multimodal benchmark function, composed of a quadratic convex function and an oscillatory nonconvex function. The comparative importance of Griewank’s two major parts alters in different dimensions. Different from most test functions, an unusual phenomenon appears when optimizing the Griewank function. The Griewank function first becomes more difficult and then becomes easier to optimize with the increase of dimension. In this study, from the methodology perspective, this phenomenon is explained by structural, mathematical, and quantum analyses. Furthermore, frequency transformation and amplitude transformation are implemented on the Griewank function to make a generalization. The multi-scale quantum harmonic oscillator algorithm (MQHOA) with quantum tunnel effect is used to verify its characteristics. Experimental results indicate that the Griewank function’s two-scale structure is the main reason for this phenomenon. The quantum tunneling mechanism mentioned in this paper is an effective method which can be generalized to analyze the generation and variation of solutions for numerous swarm optimization algorithms.

A Binary Multi-Scale Quantum Harmonic Oscillator Algorithm for 0–1 Knapsack Problem With Genetic Operator

The 0–1 knapsack problem is a typical discrete combinatorial optimization problem with numerous applications. In this paper, a binary multi-scale quantum harmonic oscillator algorithm (BMQHOA) with genetic operator is proposed for solving 0–1 knapsack problem. The framework of BMQHOA is consisted of three nested phases including energy level stablization, energy level decline and scale adjustment. In BMQHOA, the number of different bits between solutions is defined as the distance between solutions to map the continuous search space into the discrete search space. Repair operator with greedy strategy is adopted in BMQHOA to guarantee the knapsack capacity constraint. The current best solution is used to perform a random mutation with the origin solutions, making solutions evolve towards the current optimal solution. The performance of BMQHOA is evaluated on two low-dimensional and three high-dimensional KP01 data sets, and computational results are compared with several state-of-art 0–1 knapsack algorithms. Experimental results demonstrate that the proposed BMQHOA can get the best solutions of most knapsack data sets, and performs well on convergence and stability.

Multiscale Quantum Harmonic Oscillator Algorithm With Multi-Harmonic Oscillators for Numerical Optimization

The multiscale quantum harmonic oscillator algorithm (MQHOA) is inspired by the physical meaning of quantum wavefunction. Particles have no efficient interaction and share a wavefunction during the evolution process. In this paper, a multi-harmonic oscillator strategy is introduced, which employs multiple wavefunctions to generate new particles. The external population information can be utilized in the process of evolution. The proposed method enhances the cooperation and interaction for particles by a local collaborative operation. Moreover, an adaptive weight operator is employed in the proposed algorithm, which makes a fine-tune of solutions to keep the diversity of particles. The proposed algorithm is verified on standard benchmark functions. Wilcoxon rank sum test is adopted to ascertain the superiority of the proposed algorithm. The experiments have been conducted with several renowned heuristic algorithms. The numerical results reveal that the proposed algorithm outperforms the comparison algorithms for numerical optimization.

Multiscale Quantum Harmonic Oscillator Algorithm With Strict Metastability Constraints for Multi-Modal Optimization

Multi-modal optimization is a troublesome problem faced by optimization algorithms. The multiscale quantum harmonic oscillator algorithm (MQHOA) utilizes group statistics strategy to evaluate the state of the population and neglects the individual state. It will lead the particles to be trapped in local optima when addressing multi-modal optimization problems. This paper proposes a modified MQHOA by introducing strict metastability constraints strategy (MQHOA-SMC). The new strategy adopts a joint constraint mechanism to make the particle states mutual constraint with each other. The modified algorithm enhances the ability to find a better quality solution in local areas. To demonstrate the efficiency and effectiveness of the proposed algorithm, simulations are carried out with SPSO2011, ABC, and QPSO on classical benchmark functions and with the newly CEC2013 test suite, respectively. The computational results demonstrate that MQHOA-SMC is a competitive algorithm for multi-modal problems.

Multi-Scale Quantum Harmonic Oscillator Algorithm With Truncated Mean Stabilization Strategy for Global Numerical Optimization Problems

A multi-scale quantum harmonic oscillator algorithm (MQHOA) is a quantum population-based algorithm proposed recently. It utilizes the quantum wave function to locate the global optimum of a global numerical optimization problem. As the MQHOA employs the elitism to replace the worst particle in each iteration cycle, it reduces one of the particles in each run, which will cripple the diversity of the population and slow down the convergence speed. Therefore, the particles will be easily trapped into local optima. In this paper, we suggest a new MQHOA with truncated mean stabilization (TS-MQHOA) policy to alleviate the above-mentioned problems. The theoretical and experimental analyses indicate that the truncated mean stabilization strategy helps to diversify the populations and improve the convergence efficiency. The proposed TS-MQHOA is evaluated on a number of dimensionwise unimodal and multimodal CEC benchmark functions, and the computational results are compared with several popular population-based algorithms. The experimental results on complex test problems demonstrate that the proposed TS-MQHOA, in most function evaluations, is able to obtain better convergence toward the global optimum compared with several renowned heuristic algorithms based on swarm intelligence. Meanwhile, the comparative results reveal the competitiveness and superiority of the proposed algorithm, especially on high-dimensional function evaluations.

Analysis of Multiscale Quantum Harmonic Oscillator Algorithm Based on a New Multimode Objective Function

The wavefunction is an important element of the multiscale quantum harmonic oscillator algorithm (MQHOA). To verify the physical model and the multimode optimization performance of the MQHOA for multimode optimization in this paper, we define a multidimensional harmonic-Gaussian potential function. When the wavefunction of the MQHOA for multimode optimization and the probability amplitude of an ammonia molecule are compared, the ground-state wavefunction can reflect the probability distribution of the two-state ammonia molecule. We obtain the extrema of the proposed function using the Hessian matrix and optimize the proposed function with the multimode algorithm. The experiments show that the multimode algorithm can determine the extrema of the proposed function with appropriate parameters. Changes in the proposed function barriers have little effect on the optimization ability of the multimode algorithm. The optimization ability of the multimode algorithm is determined by its own wavefunction.

Multi-Scale Quantum Harmonic Oscillator Algorithm With Centroid Motion for Global Numerical Optimization Problems

This paper proposes a new approach for function optimization using a new variant of multi-scale quantum harmonic optimization algorithm (MQHOA). The new approach introduces a centroid motion to improve the convergence efficiency, which is called MQHOA with centroid motion (CM-MQHOA). Instead of replacing the worst particle by the current best individual in the quantum harmonic oscillator process in MQHOA, the weakest player is replaced by a current centroid position in the proposed algorithm. Simple mechanisms are added to maintain the diversity of the population and help achieve the global optima in difficult unimodal and multimodal search spaces. The benefits of the proposed algorithm are improved performance in terms of effectiveness, reliability, accuracy, and efficiency. The approach appears to be able to efficiently deal with several unimodal and multimodal benchmark functions. A variety of standard benchmark functions are used to illustrate the proposed approach. The experimental results are compared with several state-of-the-art optimization algorithms. The comparative results indicate the competitiveness of the proposed algorithm and suggest a viable and attractive addition to the portfolio of computational intelligence techniques.

Multi-scale quantum harmonic oscillator algorithm for global numerical optimization

This paper aims to provide a novel metaheuristic algorithm motivated from quantum motion entitled Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA). The calculation accuracy of MQHOA is adjustable. The physical model and mathematical analysis of the proposed algorithm are detailed and well interpreted in this paper. The structure of MQHOA is very simple, including merely two phases, the quantum harmonic oscillator process (QHO process) and multi-scale process (M process). Experiments are carried out to validate the effectiveness and efficiency of MQHOA by applying it to 30 well defined benchmark functions. We also compare MQHOA with several well-known metaheuristic algorithms, such as genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO) and quantum particle swarm optimization (QPSO). The comparative results indicate the competitive and superior performance of the proposed algorithm in both convergence speed and optimal solution accuracy.

Multiscale Quantum Harmonic Oscillator Algorithm for Multimodal Optimization.

This paper presents a variant of multiscale quantum harmonic oscillator algorithm for multimodal optimization named MQHOA-MMO. MQHOA-MMO has only two main iterative processes: quantum harmonic oscillator process and multiscale process. In the two iterations, MQHOA-MMO only does one thing: sampling according to the wave function at different scales. A set of benchmark test functions including some challenging functions are used to test the performance of MQHOA-MMO. Experimental results demonstrate good performance of MQHOA-MMO in solving multimodal function optimization problems. For the 12 test functions, all of the global peaks can be found without being trapped in a local optimum, and MQHOA-MMO converges within 10 iterations.