A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Advanced Search
Volume 7 Issue 3
Apr.  2020

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 11.8, Top 4% (SCI Q1)
    CiteScore: 17.6, Top 3% (Q1)
    Google Scholar h5-index: 77， TOP 5
Turn off MathJax
Article Contents
Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2020  >  7(3): 901-909
Xiaoli Yin, Chunming Li and Yuan Zhang, "Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method," IEEE/CAA J. Autom. Sinica, vol. 7, no. 3, pp. 901-909, May 2020. doi: 10.1109/JAS.2019.1911735
Citation: Xiaoli Yin, Chunming Li and Yuan Zhang, "Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method," IEEE/CAA J. Autom. Sinica, vol. 7, no. 3, pp. 901-909, May 2020. doi: 10.1109/JAS.2019.1911735
shu

Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method

doi: 10.1109/JAS.2019.1911735
Funds:  This work was supported in part by the Teaching Reform Research Foundation of Shengli College in China University of Petroleum (East China) (JG201725), the Natural Science Foundation Shandong Province of China (ZR2018PEE009), and the Project of Science and Technology of Shandong Universities in China (J17KA044, J17KB061)
More Information
    Abstract
  • Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems’ objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843% of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360°, because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05°. Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71% less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained, but does not require derivability and does not take into account truncation error and rounding error. Thus, the application scopes of many optimization methods are extended.

     

    • FullText(HTML)
  • loading
    • References(32)
  • [1]
    H. Mu, Z. Li, W. Huo, and Z. Jin, “Hash map optimization and its application in column-oriented database query,” Chinese J. Frontiers of Computer Science and Technology, vol. 10, no. 9, pp. 1250–1261, 2016.
    [2]
    Z. Jin, H. Yang, and G. Yin, “A numerical approach to optimal dividend policies with capital injections and transaction costs,” Acta Mathematicae Applicatae Sinica (English Serie), vol. 33, no. 1, pp. 221–238, 2017. doi: 10.1007/s10255-017-0653-6
    [3]
    J. Qian, D. Wang, X. Miao, and G. Su, “Numerical simulation and optimization of automobile outflow field based on CFD,” Chinese Manufacturing Automation, vol. 38, no. 4, pp. 74–76, 2016.
    [4]
    J. Lin, L. Guan, J. Han, and M. Lu, “Design and optimization of the parallel elliptical vibration cutting mechanism,” Chinese J. Machine Design, vol. 35, no. 5, pp. 53–59, 2018.
    [5]
    J. Shi, L. Wang, C. Luo, G. Cai, C. Shen, and Z. Zhu, “Sparse repersentation for gearbox compound fault feature extraction based on majorization-minimization algorithm,” Chinese J. Vibration Engineering, vol. 30, no. 6, pp. 1045–1055, 2017.
    [6]
    S. Zou, I. Kokyu, I. Norio, and F. Yuji, “Design optimization of friction damper with coupling mechanism for seismic response of base isolated structure,” Chinese J. Vibration Engineering, vol. 29, no. 2, pp. 201–206, 2016.
    [7]
    Y. Hu and X. Liu, “Optimization of grouping evacuation strategy in high-rise building fires based on graph theory and computational experiments,” IEEE/CAA J. Autom. Sinica, vol. 5, no. 6, pp. 1104–1112, 2018. doi: 10.1109/JAS.2018.7511231
    [8]
    X. Bi, L. Zhang, and J. Xiao, “Constrained multi-objective optimization algorithm based on dual populations,” Chinese J. Computer Research and Development, vol. 52, no. 12, pp. 2813–2823, 2015.
    [9]
    X. Guo, L. Lu, and C. Zhu, “Two phase many-objective optimization algorithm based on Pareto dominance relationship,” Chinese J. Frontiers of Computer Science and Technology, vol. 12, no. 8, pp. 1350–1360, 2018.
    [10]
    J. Li, H, L i, and X. Zhao, “Multi-objective optimization of hybrid electrical vehicle based on immune genetic algorithm,” Computer Engineering and Applications in China, vol. 54, no. 4, pp. 237–262, 2018.
    [11]
    L. Wang, L. Qiao, and L. Wei, “Optimal convolutional neural networks learning method combined with genetic algorithm,” Computer Engineering and Design in China, vol. 38, no. 7, pp. 1945–1950, 2017.
    [12]
    J. Xie, S. Su, and J. Wang, “Search strategy of artificial bee colony algorithm guided by approximate gradient,” Chinese J. Frontiers of Computer Science and Technology, vol. 10, no. 12, pp. 1773–1782, 2016.
    [13]
    Y. Ke and C. Ma, “A preconditioner for elliptic PDE-constrained optimization problems,” Mathematic Numerica Sinica in China, vol. 39, no. 1, pp. 70–80, 2017.
    [14]
    C. L. Zhang, “The improvement of Newton method and the validation of optimization idea of blind walking repeatedly”, Advanced Materials Research, vol. 1270, no. 503, pp. 4061–4064, 2011.
    [15]
    C. Li, H. Wang, W. Xu, and X. Li, “Multidimensional two-time fitting function optimization method,” J. Gansu Sciences, vol. 29, no. 5, pp. 26–28, 2017. doi: 10.16468/j.cnki.issn1004-0366.2017.05.006
    [16]
    Q. Liu, X. Liu, X. Yin, and C. Li, “Conjugate method of adjacent directions based on gradient vector of objective function,” J. Gansu Sciences, vol. 29, no. 5, pp. 15–21, 2017. doi: 10.16468/j.cnki.issn1004-0366.2017.05.004
    [17]
    X. Jiang and J. Jian, “A self-adjusting Polak-Ribiere-Polyak type conjugate gradient method,” Acta Mathematicae Applicatae Sinica, vol. 40, no. 3, pp. 449–460, 2017.
    [18]
    X. L. Yin, F. Sun, and C. M. Li, “Six search optimization method on obtaining conjugate direction after continuous negative gradient directions,” Chinese J. Frontiers of Computer Science and Technology, vol. 13, no. 9, pp. 1604–1612, 2019.
    [19]
    X. L. Yin, F. Sun, and C. M. Li, “Half-step constrained mechanical optimization method based on conjugate direction,” J. Gansu Sciences, vol. 30, no. 5, pp. 29–36, 2018. doi: 10.16468/j.cnki.issn1004-0366.2018.06.006
    [20]
    J. Chen and L. Zhang, “Numerical approximation of solution for the coupled nonlinear Schrödinger equations,” Acta Mathematicae Applicatae Sinica (English Serie), vol. 33, no. 2, pp. 435–450, 2017. doi: 10.1007/s10255-017-0672-3
    [21]
    X. Li, Q. Han, J. Wang, Q. Wang, and X. Duan, “ERT image reconstruction based on improved CG method,” Chinese J. Scientific Instrument, vol. 37, no. 7, pp. 1673–1679, 2016.
    [22]
    J. Li, X. Lv, T. Tao, and M. Lai, “Conditional gradient algorithm for solving operator equation least squares problem under the bound constraints,” Mathematica Numerica Sinica, vol. 38, no. 4, pp. 372–390, 2016.
    [23]
    J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, 1983.
    [24]
    O. A. Arqub and Z. Abo-Hammour, “Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm,” Information Sciences, vol. 2014, no. 279, pp. 396–415, 2014.
    [25]
    T. F. Coleman and J. G. More, “Estimation of sparse Jacobean matrices and graph coloring problems”, Report No. ANL-81-39, Argonne National Laboratory, Argonne, Ill., 1981.
    [26]
    P. N. Brown and Y. Saad, “Hybrid Krylov methods for nonlinear systems of equations,” SIAM J. Scientific and Statistical Computing, vol. 11, no. 3, pp. 450–481, 1990. doi: 10.1137/0911026
    [27]
    W. Cheney and D. Kincaid, Linear Algebra: Theory and Applications, Sudbury, MA: Jones and Bartlett. 2009, pp. 544–558. ISBN 978-0-7637-5020-6.
    [28]
    L. Pursell and S. Y. Trimble, “Gram-Schmidt orthogonalization by Gauss elimination,” The American Mathematical Monthly, vol. 98, no. 6, pp. 544–549, 1991. doi: 10.2307/2324877.Retrieved5March2017
    [29]
    E. D. Solomentsev, Euclidean Space, Encyclopedia of Mathematics, 2011, Springer Retrieved, 2014.
    [30]
    C. Li, “Negative gradient direction method for blind person exploring the way,” J. Gansu Sciences, vol. 28, no. 5, pp. 116–122, 2016. doi: 10.16468/j.cnki.issn1004-0366.2016.05.026
    [31]
    C. Li, “Blind-walking optimization method,” J. Networks, vol. 5, no. 12, pp. 1458–1466, 2010.
    [32]
    X. Liu, X. Yin, and C. Li, “On the improvement of the Powell mechanical optimization method,” Chinese J. Machine Design, vol. 36, no. 6, pp. 80–86, 2019.
    • Related Articles
    • Supplements(0)

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(5)

    Get Citation
    PDF
    XML

    Article Metrics

    Article views (1130) PDF downloads(40) Cited by()

    Highlights

    • Numerical mixed partial derivative based on quadratic hypothesis.
    • Numerical results verification by coordinate rotation.
    • Numerical optimization method and general optimization method.

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return