Stochastic Matrix-Free Equilibration

Steven Diamond () and Stephen Boyd ()
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Steven Diamond: Stanford University
Stephen Boyd: Stanford University

Journal of Optimization Theory and Applications, 2017, vol. 172, issue 2, No 5, 436-454

Abstract: 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.

Keywords: Matrix balancing; Preconditioning; Convex optimization; Stochastic optimization; 65F35 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-016-0990-2

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