Exact algorithms for linear matrix inequalities

TL;DR

This paper introduces an exact algorithm for computing points in spectrahedra defined by linear matrix inequalities, with complexity bounds and practical improvements over existing methods.

Contribution

The paper presents a novel exact algorithm for spectrahedron points, handling cases without interior points and providing algebraic representations with complexity analysis.

Findings

Algorithm computes an exact point in the spectrahedron or determines emptiness.

Complexity is quadratic in a multilinear Bézout bound, polynomial when one dimension is fixed.

Experimental results show practical improvements over existing algebraic algorithms.

Abstract

Let $A(x)=A\_0+x\_1A\_1+...+x\_nA\_n$ be a linear matrix, or pencil, generated by given symmetric matrices $A\_0,A\_1,...,A\_n$ of size $m$ with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI). We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree $d$ of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear B\'ezout bound of $d$. When $m$ (resp. $n$) is fixed, such a bound, and hence the complexity, is polynomial in $n$ (resp. $m$). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms.