Dualities in Convex Algebraic Geometry

TL;DR

This paper explores the duality concepts in convex algebraic geometry, linking optimization, algebraic geometry, and semidefinite programming, and reveals how polynomial program solutions relate to algebraic dual hypersurfaces.

Contribution

It compares different duality notions in convex algebraic geometry and connects the optimal value of polynomial programs to algebraic dual hypersurfaces, advancing theoretical understanding.

Findings

Optimal polynomial program value is an algebraic function.

Minimal polynomial is given by the hypersurface projectively dual to the constraint set.

Boundary structure of convex hulls relates to spectrahedral shadows.

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.