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Complexity Analysis of a Sampling-Based Interior Point Method for Convex Optimization

Author

Listed:
  • Riley Badenbroek

    (Econometric Institute, Erasmus University Rotterdam, Rotterdam 3062 PA, Netherlands)

  • Etienne de Klerk

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg 5037 AB, Netherlands)

Abstract

We develop a short-step interior point method to optimize a linear function over a convex body assuming that one only knows a membership oracle for this body. The approach is based a sketch of a universal interior point method using the so-called entropic barrier. It is well known that the gradient and Hessian of the entropic barrier can be approximated by sampling from Boltzmann-Gibbs distributions and the entropic barrier was shown to be self-concordant. The analysis of our algorithm uses properties of the entropic barrier, mixing times for hit-and-run random walks, approximation quality guarantees for the mean and covariance of a log-concave distribution, and results on inexact Newton-type methods.

Suggested Citation

  • Riley Badenbroek & Etienne de Klerk, 2022. "Complexity Analysis of a Sampling-Based Interior Point Method for Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 779-811, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:779-811
    DOI: 10.1287/moor.2021.1150
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