Dualities in Convex Algebraic Geometry
Philipp Rostalski and Bernd Sturmfels
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.
中文速览
多种对偶理论表面上各自独立，但在多项式优化与半定规划（semidefinite programming）的交汇地带，它们其实是同一件事的不同侧面——这正是本文要厘清的核心问题。作者系统比较了三种对偶：凸体对偶、射影代数簇对偶（projective duality）、以及优化中的拉格朗日/KKT对偶，并证明多项式规划最优值函数的极小多项式，恰好由约束集的射影对偶超曲面（projectively dual hypersurface）的方程给出。文章还梳理了紧代数簇凸包边界结构的最新成果，将其与谱面影（spectrahedral shadow）的Lasserre表示做了对比，并以三维"枕头"形谱面体（spectrahedron）为贯穿全文的具体例子，把抽象理论落地为可计算的代数方程。这项工作为凸代数几何这一新兴领域提供了统一的理论框架，揭示了代数几何中经典的射影对偶如何在实际优化算法的设计与分析中发挥核心作用。
原文 arXiv:1006.4894;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1006.4894v1