“Powell's method”的意思、由来-开放百科全书

词条

Powell's method

释义

References

Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function. The function need not be differentiable, and no derivatives are taken.

The function must be a real-valued function of a fixed number of real-valued inputs. The caller passes in the initial point. The caller also passes in a set of initial search vectors. Typically N search vectors (say) are passed in which are simply the normals aligned to each axis.^[1]

The method minimises the function by a bi-directional search along each search vector, in turn. The bi-directional line search along each search vector can be done by Golden-section search or Brent's method. Let the minima found during each bi-directional line search be , where is the initial starting point and is the scalar determined during bi-directional search along . The new position () can then be expressed as a linear combination of the search vectors i.e. . The new displacement vector () becomes a new search vector, and is added to the end of the search vector list. Meanwhile, the search vector which contributed most to the new direction, i.e. the one which was most successful (), is deleted from the search vector list. The new set of N search vectors is .The algorithm iterates an arbitrary number of times until no significant improvement is made.^[1]

The method is useful for calculating the local minimum of a continuous but complex function, especially one without an underlying mathematical definition, because it is not necessary to take derivatives. The basic algorithm is simple; the complexity is in the linear searches along the search vectors, which can be achieved via Brent's method.

References

1. ^¹{{cite web|last1=Mathews|first1=John H.|title=Module for Powell Search Method for a Minimum|url=http://mathfaculty.fullerton.edu/mathews/n2003/PowellMethodMod.html|website=California State University, Fullertron|accessdate=16 June 2017}}

{{cite journal | last=Powell | first=M. J. D. | year=1964 | title=An efficient method for finding the minimum of a function of several variables without calculating derivatives | journal=Computer Journal | volume=7 | issue=2 | pages=155–162| doi=10.1093/comjnl/7.2.155 }}
{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 10.7. Direction Set (Powell's) Methods in Multidimensions | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=509}}
{{Cite book | last=Brent | first=Richard P. | year=1973 | title=Algorithms for minimization without derivatives | publisher=Prentice-Hall | publication-place=Englewood Cliffs, N.J. | isbn=0-486-41998-3 | chapter=Section 7.3: Powell's algorithm | url=https://books.google.com/books?id=6Ay2biHG-GEC&lpg=PP1}}

1 : Optimization algorithms and methods

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。