An OnL -iteration homogeneous and self-dual linear programming algorithm
Mathematics of Operations Research
Some perturbation theory for linear programming
Mathematical Programming: Series A and B
Linear programming, complexity theory and elementary functional analysis
Mathematical Programming: Series A and B
Understanding the Geometry of Infeasible Perturbations of a Conic Linear System
SIAM Journal on Optimization
Condition-Based Complexity of Convex Optimization in Conic Linear Form via the Ellipsoid Algorithm
SIAM Journal on Optimization
Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization
SIAM Journal on Optimization
On the Complexity of Computing Estimates of Condition Measures of a Conic Linear System
Mathematics of Operations Research
Complexity of convex optimization using geometry-based measures and a reference point
Mathematical Programming: Series A and B
On the behavior of the homogeneous self-dual model for conic convex optimization
Mathematical Programming: Series A and B
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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 second-order 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 Peña'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 = 0.675), 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.