  The Zariski closure of the central path which interior point algorithms track
in convex optimization problems such as linear, quadratic, and semidefinite
programs is an algebraic curve. The degree of this curve has been studied in
relation to the complexity of these interior point algorithms, and for linear
programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show
that the degree of the central curve for generic semidefinite programs is equal
to the maximum likelihood degree of linear concentration models. New results
from the intersection theory of the space of complete quadrics imply that this
is a polynomial in the size of semidefinite matrices with degree equal to the
number of constraints. Besides its degree we explore the arithmetic genus of
the same curve. We also compute the degree of the central curve for generic
linear programs with different techniques which extend to bounding the same
degree for generic quadratic programs.

Optimization and Control;Algebraic Geometry