A General Formula for the Algebraic Degree in Semidefinite Programming Hans-christian Graf Von Bothmer and Kristian Ranestad

In this note, we use a natural desingularization of the conormal variety of the variety of (n × n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming. 1. The algebraic degree in semidefinite programming Let P be a general projective space of symmetric (n×n)−matrices up to scalar multiples, and let Yr ⊂ P m be the subvariety of matrices of rank at most r. In this note we find a formula for the degree δ(m,n, r) of the dual variety Y ∗ r whenever this is a hypersurface, i.e. δ(m,n, r) is the number of hyperplanes in P in a general pencil that are tangent to Yr at some smooth point. In [3] the algebraic degree of semi-definite programming is introduced and is shown to coincide with a degree of a dual variety as above [3, Theorem 5.2]. In particular, the number δ(m,n, r) is a measure of the algebraic complexity of the rank r solution for a general objective function optimized over a generic m-dimensional affine space of symmetric (n× n)−matrices. Our general formula for δ(m,n, r) extends the results of [3]. It is expressed in terms of a function on subsequences of {1, ..., n}. Let ψi = 2 , ψi,j = j−1