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Mathematical Programming

, Volume 109, Issue 2–3, pp 445–475 | Cite as

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

  • Robert M. Freund
  • Fernando Ordóñez
  • Kim-Chuan Toh
FULL LENGTH PAPER

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.

Keywords

Behavioral measure Condition number Degeneracy Complementarity Interior-point method Semi-definite programming 

Mathematics Subject Classification (2000)

90-04 90C22 90C60 90C51 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Robert M. Freund
    • 1
  • Fernando Ordóñez
    • 2
  • Kim-Chuan Toh
    • 3
    • 4
  1. 1.MIT Sloan School of ManagementCambridgeUSA
  2. 2.Industrial and Systems EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Department of MathematicsNational University of SingaporeSingaporeSingapore
  4. 4.Singapore-MIT AllianceSingaporeSingapore

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