Exact algorithms for semidefinite programs with degenerate feasible set. (May 2021)

• Record Type: Journal Article
• Authors: Henrion, Didier; Naldi, Simone; Safey El Din, Mohab
• Is Part Of: Journal of symbolic computation. Volume 104(2021)
• Page Start: 942
• Page End: 959
• Publication Date: 2021-05
• Subjects: Semidefinite programming -- Polynomial optimization -- Exact computation -- Homotopy; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.11.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Given symmetric matrices A 0, A 1, …, A n of size m with rational entries, the set of real vectors x = ( x 1, …, x n ) such that the matrix A 0 + x 1 A 1 + ⋯ + x n A n has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. 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 semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. 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. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed.