The Algebraic Degree of Semidefinite Programming

TL;DR

This paper investigates the algebraic degree of solutions in generic semidefinite programs, linking algebraic properties to geometric structures using advanced algebraic geometry techniques.

Contribution

It introduces a novel algebraic geometric approach to determine the degree of minimal polynomials of optimal solutions in semidefinite programming.

Findings

Degree counts critical points on rank loci

Uses projective duality and determinantal varieties

Provides a geometric framework for algebraic complexity

Abstract

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.