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Positive definite matrix approximation with condition number constraint

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Abstract

Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one when we use a unitary similarity invariant norm as a metric. We can especially convert it to a univariate piecewise convex optimization problem when we use the Ky Fan p-k norm. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.

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Acknowledgments

The second author was supported by Grant-in-Aid for Young Scientists (B) 22710136.

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Authors and Affiliations

  1. 2-12-1-W9-60, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Mirai Tanaka & Kazuhide Nakata

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  1. Mirai Tanaka
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  2. Kazuhide Nakata
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Correspondence to Mirai Tanaka.

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Tanaka, M., Nakata, K. Positive definite matrix approximation with condition number constraint. Optim Lett 8, 939–947 (2014). https://doi.org/10.1007/s11590-013-0632-7

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  • DOI: https://doi.org/10.1007/s11590-013-0632-7

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