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

Yan HUANG; Jian-ping LI; Peng WANG

doi:10.1631/FITEE.1900155

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

Regular Papers | Updated：2022-05-19

- Unusual phenomenon of optimizing the Griewank function with the increase of dimension
- Griewank函数优化过程中的独特现象研究
- Frontiers of Information Technology & Electronic Engineering Vol. 20, Issue 10, Pages: 1344-1360(2019)
- Affiliations：
  
  School of Computer Science and Technology, Huaiyin Normal University, Huaioan 223000, China
  School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
  Jiangsu Key Laboratory of Media Design and Software Technology, Jiangnan University, Wuxi 214122, China
  School of Computer Science and Technology, Southwest Minzu University, Chengdu 610225, China
- Author bio：
  
  [ "Yan HUANG, E-mail: hep128@qq.com" ]
  Jian-ping LI, jpli2222@uestc.edu.cn
  [ "Peng WANG, qhoalab@163.com" ]
- Funds：
- DOI：10.1631/FITEE.1900155
  CLC： TP3-05
- Received：19 March 2019，
  
  Revised：2019-10-10，
  
  Published：2019-10
- Accepted：
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Yan HUANG, Jian-ping LI, Peng WANG. Unusual phenomenon of optimizing the Griewank function with the increase of dimension[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(10): 1344-1360.
DOI：

Yan HUANG, Jian-ping LI, Peng WANG. Unusual phenomenon of optimizing the Griewank function with the increase of dimension[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(10): 1344-1360. DOI： 10.1631/FITEE.1900155.

摘要

Griewank函数是一类由二次凸函数和振荡非凸函数构成的典型多模测试函数。这两个组成部分在不同维数下显示出不同的相对重要性。不同于其他多数测试函数，随着函数维数增加，Griewank函数在优化过程出现优化难度先变难、后变易的现象。本文首先通过结构分析、数学分析和量子分析，从方法论角度解释该现象。然后，通过频率变换和幅度变换对Griewank函数作一般化处理。运用具有量子隧道效应的多尺度量子谐振子算法验证Griewank函数特性。实验结果表明Griewank函数的双尺度结构是该现象的主要原因。本文所提量子隧道效应可用于多种群体优化算法中分析解的生成和变化。

Abstract

The Griewank function is a typical multimodal benchmark function

composed of a quadratic convex function and an oscillatory nonconvex function. The comparative importance of Griewankos 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 functionos 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.

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