The degree of the central curve in semidefinite, linear, and quadratic programming

TL;DR

This paper investigates the algebraic properties of the central path in convex optimization, revealing its degree and genus for various programs, and connecting it to algebraic geometry concepts like maximum likelihood degree.

Contribution

It establishes a new link between the degree of the central curve in semidefinite programs and the maximum likelihood degree, providing explicit formulas and bounds.

Findings

Degree of the central curve in semidefinite programs equals the maximum likelihood degree.

The degree is a polynomial in matrix size with degree equal to the number of constraints.

Results extend to bounds for quadratic and linear programs.

Abstract

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation to the complexity of these interior point algorithms, and for linear programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. New results from the intersection theory of the space of complete quadrics imply that this is a polynomial in the size of semidefinite matrices with degree equal to the number of constraints. Besides its degree we explore the arithmetic genus of the same curve. We also compute the degree of the central curve for generic linear programs with different techniques which extend to bounding the same degree for generic quadratic programs.