On computing the minima of quadratic forms (Preliminary Report)

A. Yao
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引用次数: 6

Abstract

The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) = (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.
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二次型最小值的计算(初步报告)
最近,c . William Gear[1]提出了以下问题:设F(x1,x2,…,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c是n变量的二次型。我们希望通过一系列自适应函数求值来计算点x→(0)= (x1(0),…,xn(0)),在此点F达到最小值。很明显,通过计算F(x→)在1/2(n+1)(n+2)+1点处的值,我们可以确定系数a'ij,bi,c,从而找到点x→(0)。Gear的问题是,“有多少评估是必要的?”在本文中,我们将证明在最坏的情况下,任何这样的算法都需要O(n2)次求值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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