Questions tagged [linear-matrix-inequality]
Linear Matrix Inequalities (LMIs)
124
questions
1
vote
1answer
20 views
Constrained matrix equation
I have a data matrix $X \in \mathbb{R}^{n \times m}$, with $\mathrm{rank}(X) = n$ and $n \leq m$. I'm trying to understand if I can find some square matrix $C \in \mathbb{R}^{m \times m}$ such that
$$
...
1
vote
0answers
73 views
+50
Mistake in Paper about LMI characterisation of trigonometric polynomial curve?
The following is taken from Efficient Large-Scale Filter/Filterbank Design via LMI Characterization of Trigonometric Curves by Hoang Duong Tuan, Tran Thai Son, Ba-Ngu Vo, and Truong Q. Nguyen
Consider ...
12
votes
0answers
241 views
A puzzling KKT for LMI vs. scalar constraint
I am trying to understand the KKT conditions for LMI constraints in order to solve my original question in KKT conditions for $\max \log \det(X)$ with LMI constraints.
In the meantime, I found a much ...
1
vote
1answer
30 views
Reformulation of LMI
In a paper I have read, the authors reformulated the following LMI
$$X_{t} \succeq \left[\begin{array}{cc}\alpha_{i}\left(B_{t} U+C_{t}\right)^{\top} E_{i}\left(B_{t} U+C_{t}\right) & \left( B_{t} ...
3
votes
0answers
76 views
KKT conditions for $\max \log \det(X)$ with LMI constraints
I am trying to derive the KKT conditions for the following convex optimization problem where $A$ is a given matrix:
$$\begin{array}{ll} \underset{X,Y,Z}{\text{minimize}} & - \log \det \left(I + Z +...
1
vote
1answer
27 views
The equivalence of a 2 by 2 positive semidefinite matrix and a 3 by 3 positive semidefinite matrix?
I came across the following:
$$\begin{bmatrix}
-x^TAx-2b^Tx+c &-(Ax+b)^TR\\
-R(Ax+b) & \lambda I -RAR
\end{bmatrix}\geq0 \iff
\begin{bmatrix}
b^TA^{-1}b+c & 0 &(x+A^{-1}b)^T\\
0 & ...
0
votes
1answer
22 views
Linear matrix inequality given the bounds
If $A\succeq A_{\min}\succ 0$, $B\succeq B_{\min}\succ 0$, will the following be true?
$$ABA^\top \succeq AB_{\min}A^\top$$
How can I prove it?
1
vote
0answers
15 views
A problem combining linear and nonlinear matrix inequalities
I have a matrix inequality problem (don't know if this holds):
$A \in \mathbb{R}^{m \times n}$, $B_i \in \mathbb{R}^{n \times n}$, $i=1, 2, \dots, m$. $B_i$ is Hermitian. $Ax>0$ doesn't have a ...
2
votes
1answer
46 views
Is there an efficient algorithm to determine if a linear matrix inequality has a solution?
Are there any fast algorithms to determine if a linear matrix inequality (LMI) problem $Ax \leq b$ has a solution?
I am aware that linear programming and the simplex algorithm in particular may be ...
0
votes
0answers
19 views
equivalency in LMI formulation of LQR problem
In the LMI formulation of LQR problem,
solving the following optimization problem is said to be equivalent to solving the original LQR problem:
min trace(P)
sbj to
P>0
[A'PA+QāP A'PB; B'PA R+B'PB]āŖ°...
4
votes
1answer
83 views
Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?
I have the constraint
\begin{align}
X - Y A^\dagger Y^T\succ0,
\end{align}
where $A^\dagger$ is the pseudoinverse of $A\succeq0$. Can we still use the Schur complement to write the constraint as an ...
1
vote
2answers
54 views
If $A\succ0$ and $J$ is the matrix of ones, then $A \succeq J$ if and only if $\mathrm{trace}(A^{-1}J)\leq 1$
I'm trying to prove that if $A \in \mathbb{R}^{n \times n}$ is positive definite, and $J$ is the $n \times n$ matrix of ones, then $A \succeq J$ if and only if $\mathrm{trace}(A^{-1}J) = \sum_{i,j} A^{...
0
votes
1answer
18 views
Is the direction of inequality preserved when multiplying the linear matrix inequality (LMI) by a nonsingular square matrix left and right?
Let $Y$ be a square and nonsingular matrix (invertible).
Given $X \prec 0$, if we multiply $Y^*$ on the left and $Y$ on the right to the $X$ (assuming the dimension matches),
I have a claim that
$$
X \...
1
vote
0answers
37 views
Linear matrix inequality derivation from Risk-averse MPC problem
TLDR
I need to use what looks like the Schur complement to transform a linear matrix inequality but instead of a $2\times 2$ block matrix there are more blocks.
Question
I'm having trouble with a ...
1
vote
0answers
35 views
Upper bounding the largest singular value of a matrix $X$ via LMI ā is it correct?
For $z \in \Bbb C$ and $\delta > 0$, the inequality $|z| < \delta$ is equivalent to the matrix inequality
$$\begin{bmatrix} -\delta & z\\ z^* & -\delta \end{bmatrix} \prec 0$$
(Source: ...
2
votes
2answers
51 views
Why is the condition $\|Z\| < 1$ equivalent to $I - ZZ^{\top} > 0$?
As the title says, for a matrix $Z \in \mathbb{R}^{p \times q}$, the condition $\begin{Vmatrix}Z\end{Vmatrix} < 1$ equivalent to $I - ZZ^{\top} > 0$. How can I show the equivalence?
Attempt:
$\...
0
votes
0answers
20 views
Does linear matrix inequality imply eigenvalue inequality? [duplicate]
Suppose $A, B\in S^n$ and $A\succeq B$. Further suppose the eigenvalues of $A$ in non-increasing order are $\lambda_1, \lambda_2, ..., \lambda_n$, and $B$'s are $\mu_1, ..., \mu_n$. Does it imply $\...
1
vote
1answer
47 views
Matrix operator norm inequality proof
For any matrix $A \in \mathbb{R}^{m,n}$ and any $u \in \mathbb{R}^{m}$ and $v \in \mathbb{R}^n$,
how to proof the following inequality?
$$
|u^\top A v | \leq \|A\|_{op} \|u\|_1 \|v\|_1
$$
In general, ...
1
vote
1answer
46 views
Sufficient Conditions $DA^T+AD+I<0$ to Hold Based on Eigenvalues
I want to find sufficient conditions for the following matrix inequality to hold:
$$DA^T+AD+I<0$$
based on the eigenvalues of $A$ and $D$, where $D$ is diagonal with negative entries and $A$ is ...
4
votes
1answer
89 views
matrix inner product between positive semidefinite matrix and positive definite matrix
Let $F_0, F_1, \ldots, F_m$ be a $n \times n$ symmetric matrices. We define $$F(x) := F_0 + x_1 F_1 + \cdots + x_m F_m$$ Show that if there does not exist $x \in \Bbb R^m$ such that $F(x)$ is positive ...
1
vote
1answer
37 views
Inequality for the trace of the n-th power of a semi-definite matrix with trace smaller than 1.
Let $M$ be a $mxm$ positive semi-definite matrix with $tr(M)\le1$. Is there some non-trivial inequality of the type $tr(M^n)\ge f(tr(M^i), tr(M^j))$ with $f$ some function and $i,j\le n$ ?
And what ...
0
votes
0answers
32 views
Formulation of an Optimization Problem
I have the following optimization problem:
\begin{equation} \label{lip1}
\begin{aligned}
\max \lambda \ \ \ \ \text{s.t.} \\
\begin{bmatrix}
(AX+BY)^T+AX+BY+\lambda_0 X & B_w\\
* & -\mu ...
0
votes
0answers
26 views
Transformations in LMI constraints
Suppose that $T$ is a symmetric and invertible matrix. The following must be true
\begin{equation}\label{eq:SDP}\left[\begin{array}{cc}
X & A \\
A^{\top} & B
\end{array}\right] \succeq 0 \...
0
votes
0answers
29 views
Linear matrix inequality (LMI) and Schur complement
I am trying to simplify the matrix especially the element in 1by1 position.
(the blue circle in the image and the * is transposed term such as [A B|B^T C]=[A B|* C]).
Simplifying means that I want to ...
1
vote
1answer
178 views
Spectral norm - trace inequality
I am wondering whether the following is true under which assumptions on A and B?
$\operatorname{trace}(AB)\leqslant\|A\| \operatorname{trace}(B)$
The matrix norm is the spectral norm here. Maybe ...
0
votes
0answers
62 views
Reformulation to LMI constraints (H2 Feedforward Control)
I am trying to solve a problem similar to a H2 feedforward control synthesis problem with an additional constraint. I was wondering if it is possible to reformulate the problem to be a set of linear ...
3
votes
3answers
123 views
Relation between two linear matrix inequality (LMI) problems
The following $>, \geq$ signs are referred to positive definiteness.
Given an arbitrary matrix $A\in \mathbb{R}^{n\times n} $.
Problem 1: Find a matrix $P=P^\text{T}\in\mathbb{R}^{n\times n}$, $P&...
0
votes
1answer
47 views
Inequality to find the condition for positive semi-definite
I have a question regarding the convexity of the system. The formulation I arrived at has the Hessian of the following form,
$$A^TKA = \begin{bmatrix}x_1 & x_2 & ... & x_n \\ y_1 & ...
0
votes
1answer
43 views
A matrix inequality between two positive definite matrices and a positive scalar.
I am trying to prove the following inequality:
$$
P R^{-1} P - 2 \epsilon P + \epsilon^2 R \geq 0, $$
where matrices $P$ and $R$ are definite positive, and $\epsilon$ is a real-valued positive ...
0
votes
1answer
23 views
Reference to this Young Inequality for matrices
In this post:
here I saw the usage of this Young Inequality:
$$X^TY+Y^TX\leq \frac{1}{2}(X+SY)^TS^{-1}(X+SY)$$
With S any symmetric positive definite matrix. As far as I understood this could help ...
0
votes
0answers
84 views
Converting nonlinear matrix inequality to an LMI
I am fairly new to semidefinite programming (SDP). I have the following semidefinite program in matrix $X$ and vector $p$
$$\begin{array}{ll} \text{minimize} & \|Np\|^2\\ \text{subject to} & ...
0
votes
0answers
53 views
Variant of the Schur Complement
Let $$M = \left(\begin{array}{cc}{A} & {B} \\ {B^{T}} & {C}\end{array}\right)$$ be symmetric, and let $A$ be invertible. Then the Schur Complement Lemma suggests that $$C-B^{T} A^{-1} B \...
0
votes
0answers
29 views
Algorithm for LMI optimization of PSD matrix
To optimize a PSD $(n,n)$ matrix $ X $, we are asked us to propose (give ), implement and compare different Algorithms for the resolution of a Linear matrix inequality given by:
$$\begin{array}{ll} \...
0
votes
1answer
58 views
Schur complements for nonstrict inequalities
I am trying to understand the following proof from the book "Linear Matrix Inequalities in System and Control Theory".
However I am struggling to understand why $S_{2}$ must equal zero. Why isn'...
-1
votes
1answer
28 views
LMI and singular value [duplicate]
Can we have the following?
For matrices A
and B, if AāŖ°Bā¹ĻĀÆĀÆĀÆ(A)ā„ĻĀÆĀÆĀÆ(B)
?
where ĻĀÆĀÆĀÆ(ā
)
means the largest singular value.
0
votes
1answer
47 views
Singular value relation for an LMI
Can we have the following?
For matrices $A$ and $B$, if $A \succeq B \implies \overline\sigma(A) \ge \overline\sigma(B)$?
where $\overline\sigma(\cdot)$ means the largest singular value.
2
votes
1answer
71 views
How to rewrite the nonlinear matrix inequality as a linear one?
Let $t>0$ be a scalar, $P \in \mathbb{R}^{n\times n}$ is a symmetric positive-definite matrix, and $B \in \mathbb{R}^{n\times m}$ for some integers $n$ and $m$. I have the matrix inequality $tP^2-B ...
0
votes
0answers
27 views
Formulating a convex optimization problem as semidefinite program
I have the following minimization problem
$$\text{minimize} \quad f(x)= c^T F(x)^{-1} c$$
where $F : \mathbb R^n \to \mbox{Sym}_m (\mathbb R)$,
$$\mbox{dom} f = \{x \in \mathbb{R}^n \mid F(x) \succ ...
1
vote
0answers
82 views
How can we write this inequality as an LMI?
Let $X$, $Y$ and $Z$ be positive definite matrices. How can we write the following inequality as an LMI?
$$XY - Z^2 - I \succ 0$$
Here, $I$ is the identity matrix.
For example, if it was $XY-Z^2Y-I&...
1
vote
1answer
275 views
Optimization with matrix inverse constraint
I have a MILP problem. Now I want to include a new constraint to it (upper bound the diagonal elements of an inversed matrix), i.e.,
${Y(x)^{-1}}_{ii}<k_i, \forall i\in\{1,2,...,n\}$
where $Y(x)\...
1
vote
1answer
231 views
Minimizing trace of matrix inverse is an SDP?
I'm reading through this paper and on page 8 (PDF page 9), Step 2 in Algorithm 1 reads
$$
W \leftarrow \text{argmin} \{\text{tr}(WQW^T) + \alpha\, \text{tr}(Q^{ā1}): Q \succeq \epsilon\, I_n , Q_{\...
0
votes
1answer
89 views
About the use of Schur's complement. Why they are equivalent?
$$y^{\top}Qy+y^{\top}q+r\geq -ay^{\top}x-b, \qquad \forall y \in \mathbb R^{n}$$
where $Q \succeq 0$. One can use Schur's complement to replace it by an equivalent linear matrix inequality (LMI).
$$...
3
votes
0answers
65 views
LMI-based versus standard form semidefinite programs
In the context of semidefinite programming (SDP), under what conditions is it preferable to formulate and solve an LMI-based SDP rather than an equivalent standard form SDP?
I have been told that ...
1
vote
1answer
160 views
using Schur's complement and Young's inequality to reduce matrix algebraic equation to LMI
This questions concerns the practical implementation of schur's complement and Young's inequality.
Consider the following
\begin{align}
\begin{pmatrix} \begin{pmatrix}
\mathbb{A}_{\mathbb{Z}} & ...
1
vote
2answers
68 views
Finite horizon Riccati solved by LMI
Considering the system $x_{k+1}=Ax_k+Bu_k$ with quadratic cost
$J^* = \min x_N^T S x_N + \sum_{k=0}^{N-1} x_k^T Qx_k+u^T_kRu_k$
where $Q,S\succeq 0, R\succ 0$. The optimal state feedback is found ...
1
vote
2answers
85 views
Schur complement without inverted term and YALMIP solving
first of all, excuse my poor English :( Anyway, I need to factorize the equation below in terms of Schur Complement.
$\begin{equation}
{ x }^{ T }\left( PA+{ A }^{ T }P \right) x<-\gamma { x }^{ ...
2
votes
1answer
61 views
Simplest way to prove that the following constraint is convex [closed]
We have symmetric and positive semidefinite matrix variables $W_1, W_2, \dots, W_K$. Further, we have
$$Q_i = W_i - \sum_{j=1,j \neq i}^K W_j$$
and a constant real vector $x$. How to show that the ...
0
votes
0answers
54 views
Convert Discrete Riccati Equation to LMI for multi-model LQG problem
I am interested in solving the multi-model discrete-time LQG problem, that is to synthesize a stabilizing output feedback controller that stabilizes all systems with matrices $A_i, B_i,C_i,D_i$ with $...
0
votes
1answer
279 views
Maximize the smallest nonzero singular value
I want to maximize the smallest nonzero singular value of (non-square) matrix $X$. This is equivalent to maximizing $\lambda_{\min}(X^\top X)$, which can be reformulated as follows
$$\begin{array}{ll}...
1
vote
1answer
140 views
Convert the following Riccati equation into an LMI
I need help to convert the following discrete-time Riccati eqn. into an LMI with two decision variables $(K, P)$:
$$ (A+BK)^T P (A+BK) - P +Q + K^T R K \prec 0$$
The trick, at least for the ...
