[Paper Review] Positive Polynomials and Projections of Spectrahedra

João Gouveia, Tim Netzer|arXiv (Cornell University)|Nov 14, 2009

Advanced Optimization Algorithms Research15 references4 citations

TL;DR

This paper establishes a Positivstellensatz for projections of spectrahedra—sets that are not necessarily basic closed semialgebraic—by introducing a quadratic module structure that captures strictly positive polynomials on such sets. It proves that the closure of a projection of a spectrahedron is again a projection of a spectrahedron, resolves open problems on non-exposed faces, and provides a unified framework for convex hulls of curves and images of polynomial maps.

ABSTRACT

This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of the paper "Exposed faces of semidefinite representable sets" of Netzer, Plaumann and Schweighofer. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.

Motivation & Objective

• To characterize projections of spectrahedra using algebraic positivity certificates, extending known results beyond basic closed semialgebraic sets.
• To resolve open questions from prior work on non-exposed faces in spectrahedral projections.
• To establish foundational properties of projections of spectrahedra, including closure invariance.
• To unify and generalize results on convex hulls of curves and images of polynomial maps.
• To provide a constructive framework for Lasserre and theta body relaxations in polynomial optimization.

Proposed method

• Introduces a quadratic module structure, denoted $\operatorname{QM}(\mathscr{A})_d$, built from linear matrix polynomials $\mathscr{A}$, capturing sums of squares with constraints.
• Uses Carathéodory's Theorem to represent matrix polynomials as finite sums of rank-one terms, enabling finite-dimensional approximation.
• Applies the projection of spectrahedra via linear maps and intersections with linear subspaces defined by orthogonality conditions $B_i \circ M = 0$.
• Establishes that $\operatorname{QM}(\mathscr{A})_d$ lives in a finite-dimensional subspace and is the projection of a spectrahedron.
• Employs a lifting technique: a spectrahedron $\widetilde{S} \subseteq \mathbb{R}^{n+m}$ projects to $\overline{\operatorname{conv}(S)}$, enabling the construction of positivity certificates.
• Proves that the closure of any projection of a spectrahedron is again such a projection, using finite-dimensional approximation and convex hulls.

Experimental results

Research questions

• RQ1 Can a Positivstellensatz be established for projections of spectrahedra, which are not basic closed semialgebraic sets?
• RQ2 What are the limitations of Lasserre and theta body relaxation methods when applied to general semialgebraic sets and varieties?
• RQ3 Is the closure of a projection of a spectrahedron itself a projection of a spectrahedron?
• RQ4 Can the convex hull of a curve or image of a polynomial map be characterized as a projection of a spectrahedron?
• RQ5 What are the necessary and sufficient conditions for a convex set to be the projection of a spectrahedron?

Key findings

• The closure of the projection of a spectrahedron is again a projection of a spectrahedron, resolving a foundational question in convex algebraic geometry.
• A new Positivstellensatz is proven for projections of spectrahedra, showing that every strictly positive polynomial on such a set is a sum of squares plus a specific linear combination of matrix polynomial terms.
• The paper resolves open problems from [17] by constructing explicit examples of non-exposed faces in projections of spectrahedra.
• It provides a unified account of results on convex hulls of curves and images of polynomial maps, showing they are projections of spectrahedra.
• The finite-dimensional truncations $\operatorname{QM}(\mathscr{A})_d$ are shown to be projections of spectrahedra, enabling finite-dimensional optimization methods.
• The equivalence is established between a convex set being a projection of a spectrahedron and the existence of a nested, finite-dimensional, spectrahedral-projection-compatible quadratic module.