5 research outputs found
Efficiently Sampling the PSD Cone with the Metric Dikin Walk
Semi-definite programs represent a frontier of efficient computation. While
there has been much progress on semi-definite optimization, with moderate-sized
instances currently solvable in practice by the interior-point method, the
basic problem of sampling semi-definite solutions remains a formidable
challenge. The direct application of known polynomial-time algorithms for
sampling general convex bodies to semi-definite sampling leads to a
prohibitively high running time. In addition, known general methods require an
expensive rounding phase as pre-processing. Here we analyze the Dikin walk, by
first adapting it to general metrics, then devising suitable metrics for the
PSD cone with affine constraints. The resulting mixing time and per-step
complexity are considerably smaller, and by an appropriate choice of the
metric, the dependence on the number of constraints can be made
polylogarithmic. We introduce a refined notion of self-concordant matrix
functions and give rules for combining different metrics. Along the way, we
further develop the theory of interior-point methods for sampling.Comment: 54 page
Volumetric path following algorithms for linear programming
We consider the construction of small step path following algorithms using volumetric, and mixed volumetric-logarithmic, barriers. We establish quadratic convergence of a volumetric centering measure using pure Newton steps, enabling us to use relatively standard proof techniques for several subsequently needed results. Using a mixed volumetric-logarithmic barrier we obtain an O(n^1/4m^1/4L) iteration algorithm for linear programs with n variables and m inequality constraints, providing an alternative derivation for results first obtained by Vaidya and Atkinson. In addition, we show that the same iteration complexity can be attained while holding the work per iteration to O(n^2m), as opposed to O(nm^2), operations, by avoiding use of the true Hessian of the volumetric barrier. Our analysis also provides a simplified proof of self-concordancy of the volumetric and mixed volumetric-logarithmic barriers, originally due to Nesterov and Nemirovskii.