Condition-Measure Bounds on the Behavior of the Central Trajectory of a Semi-Definite Program

Manuel A. Nunez and Robert M. Freund

We present bounds on various quantities of interest regarding the central trajectory of a semi-definite program (SDP), where the bounds are functions of Renegar's condition number ${\cal C}(d)$ and other naturally-occurring quantities such as the dimensions $n$ and $m$. The condition number ${\cal C}(d)$ is defined in terms of the data instance $d=(A,b,C)$ for SDP; it is the inverse of a relative measure of the distance of the data instance to the set of ill-posed data instances, that is, data instances for which arbitrary perturbations would make the corresponding SDP either feasible or infeasible. We provide upper and lower bounds on the solutions along the central trajectory, and upper bounds on changes in solutions and objective function values along the central trajectory when the data instance is perturbed and/or when the path parameter defining the central trajectory is changed. Based on these bounds, we prove that the solutions along the central trajectory grow at most linearly and at a rate proportional to the inverse of the distance to ill-posedness, and grow at least linearly and at a rate proportional to the inverse of ${\cal C}(d)^2$, as the trajectory approaches an optimal solution to the SDP. Furthermore, the change in solutions and in objective function values along the central trajectory is at most linear in the size of the changes in the data. All such bounds involve polynomial functions of ${\cal C}(d)$, the size of the data, the distance to ill-posedness of the data, and the dimensions $n$ and $m$ of the SDP. 

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