## D. Henrion - S. Naldi - M. Safey El Din

# Exact algorithms for semidefinite programs with degenerate feasible set

created by naldi on 16 Mar 2018

modified on 18 May 2020

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BibTeX]

*Published Paper*

**Inserted:** 16 mar 2018

**Last Updated:** 18 may 2020

**Journal:** ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

**Year:** 2018

**Doi:** https://dl.acm.org/doi/proceedings/10.1145/3208976

**Links:**
Publisher page

**Abstract:**

Let $A_0, \ldots, A_n$ be $m \times m$ symmetric matrices with entries in
$\mathbb{Q}$, and let $A(x)$ be the linear pencil $A_0+x_1 A_1+\cdots+x_n A_n$,
where $x=(x_1,\ldots,x_n)$ are unknowns. The linear matrix inequality (LMI)
$A(x) \succeq 0$ defines the subset of $\mathbb{R}^n$, called spectrahedron,
containing all points $x$ such that $A(x)$ has non-negative eigenvalues. The
minimization of linear functions over spectrahedra is called semidefinite
programming (SDP). Such problems appear frequently in control theory and real
algebra, especially in the context of nonnegativity certificates for
multivariate polynomials based on sums of squares.
Numerical software for solving SDP are mostly based on the interior point
method, assuming some non-degeneracy properties such as the existence of
interior points in the admissible set. In this paper, we design an exact
algorithm based on symbolic homotopy for solving semidefinite programs without
assumptions on the feasible set, and we analyze its complexity. Because of the
exactness of the output, it cannot compete with numerical routines in practice
but we prove that solving such problems can be done in polynomial time if
either $n$ or $m$ is fixed.