On the Turing model complexity of interior point methods for semidefinite programming

TL;DR

This paper demonstrates that the short-step primal interior point method can solve semidefinite programs within fixed accuracy in polynomial time, employing Diophantine approximation to control bit-size growth during iterations.

Contribution

It proves polynomial-time complexity for interior point methods in semidefinite programming using Diophantine approximation techniques.

Findings

Interior point methods are polynomial-time for semidefinite programming.

Diophantine approximation bounds intermediate bit-sizes.

The approach extends known results for the ellipsoid method to interior point methods.

Abstract

It is known that one can solve semidefinite programs to within fixed accuracy in polynomial time using the ellipsoid method (under some assumptions). In this paper it is shown that the same holds true when one uses the short-step, primal interior point method. The main idea of the proof is to employ Diophantine approximation at each iteration to bound the intermediate bit-sizes of iterates.