A Note on the Convex Hull of Finitely Many Projections of Spectrahedra

TL;DR

This paper proves that the convex hull of finitely many projections of spectrahedra remains a projection of a spectrahedron, extending previous results to unbounded sets and broadening the understanding of semidefinitely representable sets.

Contribution

It generalizes prior work by Helton and Nie, demonstrating that the convex hull of a finite union of projections of spectrahedra is also a projection of a spectrahedron, regardless of boundedness.

Findings

Convex hull of finite union of spectrahedron projections is a spectrahedral projection

Extends Helton and Nie's bounded set result to unbounded sets

Enhances understanding of semidefinitely representable sets

Abstract

A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of a spectrahedron. This improves upon the result of Helton and Nie, who prove the same result in the case of bounded sets.

Equations

$$\left\{(x_1,\dots ,x_n)\in {\mathbb{R}}^n\mid A_0+x_1{A}_1+\cdots +x_n{A}_n⪰0\right\},$$

$$S=\left\{x\in {\mathbb{R}}^n\mid \exists z\in {\mathbb{R}}^m:A+\sum _{i=1}^n{x}_i{B}_i+\sum _{j=1}^m{z}_j{C}_j⪰0\right\},$$

$$C:=\{x\in {\mathbb{R}}^n\mid$$

$$\displaystyle\quad\bigwedge_{i=1}^{n}\left(\begin{array}[]{cc}\lambda&x_{i}\\ x_{i}&r\end{array}\right)\succeq 0\}$$

$$\mathrm{conv}(S_1\cup \cdots \cup S_t)=\left\{x\in {\mathbb{R}}^n\mid (x,1)\in K\right\},$$

$$\mathrm{conv}(S_1\cup S_2)=\{(x,y)\in {\mathbb{R}}^2\mid x>0,y>0\}\cup \{(0,0)\},$$

Full text

1. Introduction

Let $A_{0},A_{1},\ldots,A_{n}$ be real symmetric $k\times k$ matrices. The set

[TABLE]

where $\succeq 0$ means positive semidefiniteness, is called a spectrahedron. Spectrahedra are generalizations of polyhedra and occur as feasible sets for semidefinite optimization.

A projection of a spectrahedron to a subspace of $\mathbb{R}^{n}$ is often called a semidefinitely representable set. Helton and Nie [3] conjecture that every convex semialgebraic set is such a projection. See for example [1, 2, 3, 4, 5, 6, 7] for more detailed information on spectrahedra and their projections.

We prove that the convex hull of finitely many projections of spectrahedra is again a projection of a spectrahedron. This generalizes Theorem 2.2 from Helton and Nie [3], which is the same result in the case that all sets are bounded or that the convex hull is closed.

2. Result

Proposition 2.1**.**

If $S\subseteq\mathbb{R}^{n}$ is a projection of a spectrahedron, then so is $\operatorname{cc}(S)$, the conic hull of $S$.

Proof.

Since $S$ is a projection of a spectrahedron we can write

[TABLE]

with suitable real symmetric $k\times k$-matrices $A,B_{i},C_{j}$. Then with

[TABLE]

we have $C=\operatorname{cc}(S)$ (note that $C$ is a projection of a spectrahedron, since the conjunction can be eliminated, using block matrices).

To see "$\subseteq$" let some $x$ fulfill all the conditions from $C$, first with some $\lambda>0$. Then $a:=\frac{1}{\lambda}\cdot x$ belongs to $S$, using the first condition only. Since $x=\lambda\cdot a$, $x\in\operatorname{cc}(S)$. If $x$ fulfills the conditions with $\lambda=0$, then $x=0$, by the last $n$ conditions in the definition of $C$. So clearly also $x\in\operatorname{cc}(S)$.

For "$\supseteq$" take $x\in\operatorname{cc}(S)$. If $x\neq 0$ then there is some $\lambda>0$ and $a\in S$ with $x=\lambda a$. Now there is some $z\in\mathbb{R}^{m}$ with $A+\sum_{i}a_{i}B_{i}+\sum_{j}z_{j}C_{j}\succeq 0$. Multiplying this equation with $\lambda$ shows that $x$ fulfills the first condition in the definition of $C$. But since $\lambda>0$, the other conditions can clearly also be satisfied with some big enough $r$. So $x$ belongs to $C$. Finally, $x=0$ belongs to $C,$ too. ∎

Remark 2.2**.**

The additional $n$ conditions in the definition of $C$ avoid problems that could occur in the case $\lambda=0.$ This is the main difference to the approach of Helton and Nie in [3].

Corollary 2.3**.**

If $S_{1},\ldots,S_{t}\subseteq\mathbb{R}^{n}$ are projections of spectrahedra, then also the convex hull $\operatorname{conv}(S_{1}\cup\cdots\cup S_{t})$ is a projection of a spectrahedron.

Proof.

Consider $\widetilde{S}_{i}:=S_{i}\times\{1\}\subseteq\mathbb{R}^{n+1}$, and let $K_{i}$ denote the conic hull of $\widetilde{S}_{i}$ in $\mathbb{R}^{n+1}$. All $\widetilde{S}_{i}$ and therefore all $K_{i}$ are projections of spectrahedra, and thus the Minkowski sum $K:=K_{1}+\cdots+K_{t}$ is also such a projection. Now one easily checks

[TABLE]

which proves the result. ∎

Example 2.4**.**

Let $S_{1}:=\{(x,y)\in\mathbb{R}^{2}\mid x\geq 0,y\geq 0,xy\geq 1\}$ and $S_{2}=\{(0,0)\}$. Both subsets of $\mathbb{R}^{2}$ are spectrahedra, so the convex hull of their union,

[TABLE]

is a projection of a spectrahedron.