I enjoyed a series of talks by Bernd Sturmfels on some such interrelationships, which it looks like are written up in a paper by Rostalski and Sturmfels called "Dualities in Convex Algebraic Geometry."

>Abstract: Convex algebraic geometry concerns the interplay between optimization
theory and real algebraic geometry. Its objects of study include convex semialgebraic
sets that arise in semidefinite programming and from sums of squares. This article
compares three notions of duality that are relevant in these contexts: duality of convex
bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived
from Lagrange duality. We show that the optimal value of a polynomial program is an
algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary
structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of
semidefinite programming duality.