The algebraic degree of semidefinite programming

Abstract

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.

Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

“…, K d were chosen at random. This is precisely the hypothesis made by Nie et al (2009), and one of our goals is to explain the connection of Questions 1-3 to that paper.…”

“…Using notation as in (Miller and Sturmfels 2004, Def. 8.45) and (Nie et al 2009, Thm. 10), we claim that…”

“…This notation is consistent with Nie et al (2009) where this number is called the algebraic degree of semidefinite programming (SDP).…”

“…See (Nie et al 2009, §5) for basics on projective duality. We also need to compute the dual variety for its singular locus, and for the singular locus of the singular locus, etc.…”

Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

“…One may even proceed by computing the numeric SOS certificate using Newton iteration directly rather than formulating a corresponding SDP. Since the global optimum can be an algebraic number in an extension of high degree [29], we shall verify a nearby rational lower boundr r,r ∈ Q. We assume that the coefficients of the polynomials p and qj in (1) are represented exactly, for example as exact rational numbers.…”