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Stochastic Matrix-Free Equilibration

Author

Listed:
  • Steven Diamond

    (Stanford University)

  • Stephen Boyd

    (Stanford University)

Abstract

We present a novel method for approximately equilibrating a matrix using only multiplication by the matrix and its transpose. Our method is based on convex optimization and projected stochastic gradient descent, using an unbiased estimate of a gradient obtained by a randomized method. Our method provably converges in expectation and empirically gets good results with a small number of iterations. We show how the method can be applied as a preconditioner for matrix-free iterative algorithms, substantially reducing the iterations required to reach a given level of precision. We also derive a novel connection between equilibration and condition number, showing that equilibration minimizes an upper bound on the condition number over all choices of row and column scalings.

Suggested Citation

  • Steven Diamond & Stephen Boyd, 2017. "Stochastic Matrix-Free Equilibration," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 436-454, February.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:2:d:10.1007_s10957-016-0990-2
    DOI: 10.1007/s10957-016-0990-2
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    References listed on IDEAS

    as
    1. Steven Diamond & Stephen Boyd, 2016. "Matrix-Free Convex Optimization Modeling," Springer Optimization and Its Applications, in: Boris Goldengorin (ed.), Optimization and Its Applications in Control and Data Sciences, pages 221-264, Springer.
    2. Michael H. Schneider & Stavros A. Zenios, 1990. "A Comparative Study of Algorithms for Matrix Balancing," Operations Research, INFORMS, vol. 38(3), pages 439-455, June.
    3. Brendan O’Donoghue & Eric Chu & Neal Parikh & Stephen Boyd, 2016. "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1042-1068, June.
    Full references (including those not matched with items on IDEAS)

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