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

doi:10.1109/JAS.2019.1911735

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	Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method
	Xiaoli Yin; Chunming Li; Yuan Zhang
发表期刊	IEEE/CAA Journal of Automatica Sinica
ISSN	2329-9266
	2020
卷号	7 期号:3 页码:901-909
摘要	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.
关键词	Directional derivative numerical differential optimization method quadratic function hypothesis
DOI	10.1109/JAS.2019.1911735
引用统计	被引频次：1[WOS] [WOS记录] [WOS相关记录]
文献类型	期刊论文
条目标识符	http://ir.ia.ac.cn/handle/173211/42998
专题	学术期刊_IEEE/CAA Journal of Automatica Sinica
推荐引用方式 GB/T 7714	Xiaoli Yin,Chunming Li,Yuan Zhang. Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method[J]. IEEE/CAA Journal of Automatica Sinica,2020,7(3):901-909.
APA	Xiaoli Yin,Chunming Li,&Yuan Zhang.(2020).Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method.IEEE/CAA Journal of Automatica Sinica,7(3),901-909.
MLA	Xiaoli Yin,et al."Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method".IEEE/CAA Journal of Automatica Sinica 7.3(2020):901-909.

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