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A matrix generation approach for eigenvalue optimization

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Abstract

We study the extension of a column generation technique to nonpolyhedral models. In particular, we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices. At each step of the algorithm a restricted master problem in the primal space, corresponding to the relaxed dual (original) problem, is formed. A query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem. The original objective function is evaluated at the query point, and depending on its differentiability a column or a matrix is added to the restricted master problem. We discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix. The computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal-dual interior point methods on dense and sparse problems using the software SDPT3. Our numerical results illustrate that the matrix generation algorithm outperforms primal-dual interior point methods on dense problems with no structure and also on a class of sparse problems.

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Authors and Affiliations

  1. College of Business Administration, California State University San Marcos, San Marcos, California, 92096, USA

    Mohammad R. Oskoorouchi

  2. GERAD/Faculty of Management, McGill University, 1001 SherbrookeWest, Montreal, QC, H3A 1G5, Canada

    Jean-Louis Goffin

Authors
  1. Mohammad R. Oskoorouchi
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  2. Jean-Louis Goffin
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This work has been completed with the partial support of a summer grant from the College of Business Administration, California State University San Marcos, and the University Professional Development/Research and Creative Activity Grant

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Oskoorouchi, M., Goffin, JL. A matrix generation approach for eigenvalue optimization. Math. Program. 109, 155–179 (2007). https://doi.org/10.1007/s10107-006-0727-3

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  • DOI: https://doi.org/10.1007/s10107-006-0727-3

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