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On computing the minima of quadratic forms (Preliminary Report)

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Andrew Chi-Chih YaoAuthors Info & Claims

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

Pages 23 - 26

https://doi.org/10.1145/800116.803749

Published: 05 May 1975 Publication History

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Abstract

The following problem was recently raised by C. William Gear [1]: Let F(x₁,x₂,...,x_n) = Σ_i≤j a'_ijx_ix_j + Σ_i b_ix_i +c be a quadratic form in n variables. We wish to compute the point x^→(0) = (x₁⁽⁰⁾,...,x_n⁽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,b_i,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.

References

[1]

C. William Gear, private communication.

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[2]

M. O. Rabin, Solving Linear Equations by Means of Scalar Products, in Complexity of Computer Computations, edited by R. E. Miller and J. W. Thatcher, Plenum Press, 1972.

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[3]

I. R. Shafarevich, Basic Algebraic Geometry, Springer Verlag (1974), p. 71.

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Cited By

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van Apeldoorn JGilyén AGribling Sde Wolf R(2020)Convex optimization using quantum oraclesQuantum10.22331/q-2020-01-13-2204(220)Online publication date: 13-Jan-2020
https://doi.org/10.22331/q-2020-01-13-220
Möller MVuik C(2017)On the impact of quantum computing technology on future developments in high-performance scientific computingEthics and Information Technology10.1007/s10676-017-9438-019:4(253-269)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s10676-017-9438-0
Yao AAvis DRivest RHopcroft JFriedman EHarrison M(1977)An Ω(n2 log n) lower bound to the shortest paths problemProceedings of the ninth annual ACM symposium on Theory of computing10.1145/800105.803391(11-17)Online publication date: 4-May-1977
https://dl.acm.org/doi/10.1145/800105.803391

Index Terms

On computing the minima of quadratic forms (Preliminary Report)
1. Computing methodologies
  1. Artificial intelligence
    1. Knowledge representation and reasoning
  2. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Theorem proving algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Numerical differentiation
    2. Quadrature

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Published In

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

May 1975

265 pages

ISBN:9781450374194

DOI:10.1145/800116

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

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Published: 05 May 1975

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STOC '75 Paper Acceptance Rate 31 of 87 submissions, 36%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

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van Apeldoorn JGilyén AGribling Sde Wolf R(2020)Convex optimization using quantum oraclesQuantum10.22331/q-2020-01-13-2204(220)Online publication date: 13-Jan-2020
https://doi.org/10.22331/q-2020-01-13-220
Möller MVuik C(2017)On the impact of quantum computing technology on future developments in high-performance scientific computingEthics and Information Technology10.1007/s10676-017-9438-019:4(253-269)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s10676-017-9438-0
Yao AAvis DRivest RHopcroft JFriedman EHarrison M(1977)An Ω(n2 log n) lower bound to the shortest paths problemProceedings of the ninth annual ACM symposium on Theory of computing10.1145/800105.803391(11-17)Online publication date: 4-May-1977
https://dl.acm.org/doi/10.1145/800105.803391

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