On the central path of semidefinite optimization: Degree and worst-case convergence rate

TL;DR

This paper analyzes the algebraic complexity and convergence rate of the central path in semidefinite optimization using real algebraic geometry, providing bounds on degree and convergence speed.

Contribution

It introduces an algorithm for computing algebraic representations of the central path and establishes bounds on its degree and convergence rate.

Findings

Upper bound of 2^{O(m+n^2)} on the degree of the Zariski closure

Lower bound of 1/γ with γ=2^{O(m+n^2)} on convergence rate

Quantifier elimination yields explicit bounds on convergence speed

Abstract

In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bounded points in the field of algebraic Puiseux series. As a result, we derive an upper bound $2^{O(m+n^2)}$ on the degree of the Zariski closure of the central path, when $\mu$ is sufficiently small, and for the complexity of describing the limit point, where $m$ and $n$ denote the number of affine constraints and size of the symmetric matrix, respectively. Furthermore, by the application of the quantifier elimination to the real univariate representations, we provide a lower bound $1/\gamma$, with $\gamma =2^{O(m+n^2)}$, on the convergence rate of the central path.