On Two Measures of Problem Instance Complexity and Their Correlation with the Performance of SeDuMi on Second-Order Cone Problems
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OR 371-04.pdf
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Author(s)
Cai, Zhi
Freund, Robert M.
Date Issued
September 13, 2004
Publisher
Massachusetts Institute of Technology, Operations Research Center
Series/Report no.
Operations Research Center Working Paper Series;OR 371-04
Abstract
We evaluate the practical relevance of two measures of conic convex
problem complexity as applied to second-order cone problems solved using
the homogeneous self-dual (HSD) embedding model in the software
SeDuMi. The first measure we evaluate is Renegar’s data-based condition
measure C(d), and the second measure is a combined measure of the optimal
solution size and the initial infeasibility/optimality residuals denoted
by S (where the solution size is measured in a norm that is naturally
associated with the HSD model). We constructed a set of 144 secondorder
cone test problems with widely distributed values of C(d) and S
and solved these problems using SeDuMi. For each problem instance in
the test set, we also computed estimates of C(d) (using PeËœna’s method)
and computed S directly. Our computational experience indicates that
SeDuMi iteration counts and log(C(d)) are fairly highly correlated (sample
correlation R = 0.676), whereas SeDuMi iteration counts are not quite
as highly correlated with S (R = 0.600). Furthermore, the experimental
evidence indicates that the average rate of convergence of SeDuMi iterations
is affected by the condition number C(d) of the problem instance, a
phenomenon that makes some intuitive sense yet is not directly implied
by existing theory.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
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