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.

48 pages, 11 figures

References in corpus (4)

• Semidefinite Characterization and Computation of Real Radical Ideals
• Positive Polynomials and Projections of Spectrahedra
• Convex hulls of curves of genus one
• Discriminants and Nonnegative Polynomials

Cited by in corpus (9)

• Quartic Curves and Their Bitangents
• Geometry of the set of mixed quantum states: An apophatic approach
• Quantum Convex Support
• A Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints
• On real determinantal quartics
• Measures on polyhedral cones: characterizations and kinematic formulas
• Algebraic Boundaries of Convex Semi-algebraic Sets
• Semidefinite representations of non-compact convex sets
• Optimal Transport to a Variety