Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

2011/10/11 by Jiawang Nie, Nie, Jiawang
Engineering · Mathematics · #Advanced Control Systems Optimization #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Robotic Mechanisms and Dynamics #math.OC

paper · pdf · doi:10.48550/arxiv.1110.2526

11 pages, 3 figures

arxiv created 2011/10/11 · openalex publication_date 2011/10/11 · arxiv updated 2011/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V be a semialgebraic set parameterized by quadratic polynomials over a quadratic set T. This paper studies semidefinite representation of its convex hull by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that its convex hull is equal to the first order moment type semidefinite relaxation of V, up to taking closures. Similar results hold when every quadratic polynomial is homogeneous and T is defined by two homogeneous quadratic constraints,or V is defined by rational quadratic parameterizations.

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