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Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

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Abstract

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.

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Authors and Affiliations

  1. MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA, 02142, USA

    Robert M. Freund

  2. Industrial and Systems Engineering, University of Southern California, 3715 McClintock Ave, Los Angeles, CA, 90089, USA

    Fernando Ordóñez

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Singapore

    Kim-Chuan Toh

  4. Singapore-MIT Alliance, 4 Engineering Drive 3, Singapore, 117576, Singapore

    Kim-Chuan Toh

Authors
  1. Robert M. Freund
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  2. Fernando Ordóñez
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  3. Kim-Chuan Toh
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Corresponding author

Correspondence to Fernando Ordóñez.

Additional information

This paper is dedicated in memory of Jos Sturm.

This research has been partially supported through the MIT-Singapore Alliance.

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Freund, R.M., Ordóñez, F. & Toh, KC. Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems. Math. Program. 109, 445–475 (2007). https://doi.org/10.1007/s10107-006-0035-y

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  • DOI: https://doi.org/10.1007/s10107-006-0035-y

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