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Computational Intelligence and Neuroscience

Multiscale Quantum Harmonic Oscillator Algorithm for Multimodal Optimization

Peng Wang, Kun Cheng, Yan Huang, Bo Li, Xinggui Ye, Xiuhong Chen

Published Jan 1, 2018• 17 citations
Publisher

Abstract

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.

DOI
10.1155/2018/8430175

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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.
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 Oscillator Harmonic Algorithm( MQHOA) has good characteristics of global convergence and adaptability. For analyzing the specific performance of MQHOA on solution precision and speed, the comparisons of theoretical models and experiments were completed among the MQHOA, the classic Quantum Particle Swarm Optimization( QPSO) algorithm adopting the quantum-behaved model and having been widely used, and the QPSO with Random Mean best position( QPSO-RM) through solving the integer nonlinear programming problems. In simulation experiments, MQHOA achieved 100% success rate in solving seven unconstrained integer nonlinear programming problems, and was faster than QPSO and QPSO-RM in most cases. MQHOA was a little slower than QPSO and QPSO-RM in solving the two constrained integer nonlinear programming problems, but could obtain 100% success rate which was higher than the latter. Through the comparison of the convergence process of QPSO, QPSO-RM, MQHOA was faster and earlier on converging to the global optimal solution. The experimental results show that MQHOA can effectively adapt to solving the integer programming problems, and can avoid falling into the local optimal solution so as to obtain the global optimal solution. MQHOA is better than the contrast algorithms of QPSO and QPSO-RM in accuracy and convergence rate.
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.
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 dimension-wise 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.
This paper proposes a novel adaptive multi-scale quantum harmonic oscillator algorithm based on evolutionary strategies (AMQHOA-ES) for global numerical optimization. Since the original Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA) utilizes a fixed contraction factor to narrow the search scale, the searching step decreases too fast at the later stage of the evolution and is more likely to suffer premature convergence and stagnation. To improve the convergence performance, an adaptive attenuation mechanism of scaling is proposed to dynamically adjust the exploration and exploitation properties. Evolutionary strategies such as selection, crossover and DE/rand/1 mutation are implemented in the proposed algorithm to enhance the exploration and exploitation abilities. Experimental results evaluated on several unimodal and multimodal benchmark functions indicate the significant improvement of the proposed algorithm to the original MQHOA. Meanwhile, the experimental results compared with several state-of-the-art optimizers show the superiority or competitiveness of the proposed algorithm.
With the multi-scale binary sampling methods on the basis of Gaussian scaling function,the model of multi-scale quantum harmonic oscillator algorithm for high-dimensional function global optimization problems is proposed.High-dimensional function optimization process is divided into two steps,scale convergence and quantum harmonic oscillator convergence.This algorithm model is based on the same convergence process probability interpretations between function optimization problem and quantum harmonic oscillator from high-energy state to ground state.This algorithm,which has explicit physical model,can realize highdimensional function optimization without coding or complex initial conditions.Experiments and analysis are done for 15 typical two-dimensional optimized test functions and 6 typical high-dimensional optimized test functions.The results show that multi-scale quantum harmonic oscillator algorithm gets precise global optimum for high-dimensional function quickly,and withreduced frequencyapproach,the search speed of function withhigh frequency component improves significantly.