Optimization Letters

, Volume 7, Issue 6, pp 1047–1059 | Cite as

An asymptotically optimal gradient algorithm for quadratic optimization with low computational cost

  • Anatoly Zhigljavsky
  • Luc Pronzato
  • Elena Bukina
Original Paper


We consider gradient algorithms for minimizing a quadratic function in \({\mathbb{R}^n}\) with large n. We suggest a particular sequence of step-sizes and demonstrate that the resulting gradient algorithm has a convergence rate comparable with that of Conjugate Gradients and other methods based on the use of Krylov spaces. When the matrix is large and sparse, the proposed algorithm can be more efficient than the Conjugate Gradient algorithm in terms of computational cost, as k iterations of the proposed algorithm only require the computation of O(log k) inner products.


Quadratic optimization Gradient algorithms Conjugate gradient Arcsine distribution Fibonacci numbers Estimation of leading eigenvalues 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Anatoly Zhigljavsky
    • 1
  • Luc Pronzato
    • 2
  • Elena Bukina
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Laboratoire I3S, CNRS-UNSSophia AntipolisFrance

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