Questions tagged [spectrahedra]
A spectrahedron is a convex set given by the intersection of an affine space with the convex positive semidefinite cone.
16
questions
1
vote
3answers
64 views
Calculation involving determinant of a matrix
Suppose I have the following Toeplitz symmetric matrix
\begin{align}
M=\begin{bmatrix}
1 & c & c & x \\
c & 1 & c & c \\
c & c & 1 & c \\
x & c & c & 1
...
1
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0answers
23 views
Is there a way to identify singular points in a spectrahedron without finding the entire set?
I'm a physicist working in a non-linear optimization problem that reduces to a semidefinite program (SDP). In the simple low dimensional cases we workout in the research we could identify singular ...
3
votes
1answer
75 views
Geometric center of convex set of positive semidefinite matrices
Consider the set of $d\times d$ matrices that are positive semidefinite and have unit trace. This is a convex set, $S$. Is it possible to think of a geometric center of this set? The criterion for ...
25
votes
1answer
540 views
What is the surface area of the 3-dimensional elliptope?
The $n$-elliptope is defined as the set of $n$-by-$n$ correlation matrices; that is, the set of $n$-by-$n$ symmetric positive-definite matrices with ones on the diagonal. Such matrices are ...
5
votes
1answer
139 views
What is the formula for projection onto spectraplex?
A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let
$$
S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
0
votes
1answer
134 views
Finding the dual of the feasible set of a linear matrix inequality (LMI)
I am stuck at problem 3 of this homework assignment about semidefinite programming and linear matrix inequalities (LMIs).
Given a set $\mathcal{S} \subseteq \mathbb{R}^n$ that strictly contains the ...
1
vote
1answer
154 views
Does cone associated with PSD matrix always convex?
Is it true that PSD-cone always convex? (If not, please provide an example).
If this is the case, then set of PSD matrices always convex or it could happen that such set might not be convex for some ...
3
votes
2answers
199 views
Every convex polyhedron is a spectrahedron
I'm trying to show that convex polyhedra are special cases of spectrahedra. This was left as an exercise to the reader in a convex optimization text that I'm reading. I'm not sure how standard the ...
1
vote
4answers
94 views
Convexity of the set $C = \{x \in \Bbb R^n \mid x^T Ax \leq b\}$ where $A \in S^n$, $A \succeq 0$ and $b \geq 0$
I have this question in my homework and neither I nor my colleagues can solve it.
Show that the set $$C = \{x \in \Bbb R^n \mid x^T Ax \leq b\}$$ where $A$ is symmetric and positive semidefinite ...
1
vote
1answer
147 views
Why is the feasible set of solutions to an SDP a spectrahedron?
A spectrahedron is the set $$ S = \left\lbrace (x_1,\cdots,x_m)\in \mathbb{R}^m \quad|\quad A_0+ A_ix_i \succeq 0 \quad i\in [m] \right\rbrace$$
for some given symmetric matrices $A_0, A_1,\cdots,...
3
votes
3answers
261 views
Is the set $S = \{x \in [0, \infty)^2 \mid x_1 x_2^2 \leq 1\}$ convex?
We have the function $\displaystyle{y=f(x_1, x_2)=x_1\cdot x_2^2}$ and the set $S=\{x\in [0, \infty)^2 \mid f(x_1, x_2)\leq 1\}$.
I want to check if the set is convex.
$$$$
Let $x=(x_1, x_2) , y=...
19
votes
2answers
786 views
What is the volume of the $3$-dimensional elliptope?
My question
Compute the following double integral analytically
$$\int_{-1}^1 \int_{-1}^1 2 \sqrt{x^2 y^2 - x^2 - y^2 + 1} \,\, \mathrm{d} x \mathrm{d} y$$
Background
The $3$-dimensional ...
3
votes
1answer
116 views
What can we say about the form of the solution set $\{B : \mbox{tr} (AB) < 0\}$?
I have two $n \times n$ matrices $A$ and $B$, where $B$ is symmetric and p.s.d and $A$ is symmetric , rank $2$ and its two dominant eigenvalues have different signs. Considering the following ...
2
votes
2answers
2k views
How to plot the PSD cone in MATLAB
Does anybody know how I can plot in MATLAB the cone of positive semidefinite matrices as shown in the figure below? Thanks.
PSD cone
1
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1answer
272 views
A spectrahedron is a quadric cone when matrices in LMI are in $\mathbb{R}^{2\times 2}$
I am watching a lecture (just at the beginning around 0:50-0:57).
The note says when $n=2$, $\mathcal{S}$ is a quadric cone; however, it seems that the professor says "a quadratic cone".
On ...
5
votes
2answers
1k views
Solution set of an LMI is convex
I was going through Boyd and Vandenberghe's Convex Optimization book. There they mentioned (at page number $38$) that the solution set of a linear matrix inequality (LMI) is convex.
$$A(x)=x_1A_1+\...
