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Questions tagged [linear-matrix-inequality]

Linear Matrix Inequalities (LMIs)

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The minimum product of the diagonals of a PSD matrix under orthogonal transformations

Let $D = diag(d_1,\dots,d_n) \in \mathbb{R}^{n\times n}$ be a diagonal matrix such that $d_1 \geq d_2 \geq \dots \ge d_n > 0$, and let $U \in \mathbb{R}^{n\times n}$ be an arbitrary orthogonal ...
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Is the following LMI solvable? [closed]

Recently, I have been working on a control problem. The following popular LMI is very interesting: $PA+A^TP<0$, $A\in\mathbb{R}^{n\times n}$ is known and $P$ is required positive definite. It is ...
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Feasible set formed by exclusion of two convex sets

I'm working on an optimal control problem which is almost entirely composed by elements of a quadratic programming problem. The decision variable is $u \in \mathbb{U} \subseteq \mathbb{R}^2$, where $\...
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Is there any inequality involving the Frobenius norm and the dimension of matrix?

Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following: $$ \left \| AB \right \|_F \leq f(m,r,n)g(A,B) , $$ or $$ \left \| AB \...
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How can I convert a strict linear (affine) matrix inequality feasibility problem into a non-strict one?

Consider the following linear matrix inequality (LMI) feasibility problem $$ \begin{aligned} \textrm{find} \quad & X \\ \textrm{s.t.} \quad & X \succ 0 \\ \quad & F(X) \prec Q \end{aligned}...
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For a fixed positive definite $A$ anb vector $x$, is $B\mapsto x^T B^{1/2} A B^{1/2} x$ always concave?

Let $x\in R^d$ and $A\in R^{d\times d}$ positive definite. Is the map $$ B \mapsto x^T B^{1/2} A B^{1/2} x $$ always concave? One known result that gives a little hope is the Lieb inequality (cf. ...
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For a fixed positive definite $A$, is $(x,B) \to x^TB^{-1/2} A B^{-1/2} x$ always jointly convex?

Consider a real positive definite matrix $A\in R^{d\times d}$. Let $S_d^+$ be the set of symmetric positive matrices. Is it always true that $$(x,B) \to x^TB^{-1/2} A B^{-1/2} x$$ is jointly convex in ...
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Is it true that $\exists X \succeq 0, AXB \succeq 0 \implies \exists Y \succ 0, AYB \succ 0$?

Consider the rectangular matrices $A \in \mathbb R^{m \times n}$ and $B \in \mathbb R^{n \times m}$. Is it true that, if there exists a square and positive semi-definite matrix $X \succeq 0$ such that ...
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Does the solution exist for the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$? [closed]

I am studying the consensus problem of a linear multi-agent system, and a control gain needs to be designed to reach the consensus property. But the design procedure depends on the solution of the ...
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Converting nonlinear matrix inequality into an LMI

I need the following inequality to hold; $$ M+(CK)^{\text{T}}NCK > 0 $$ where $$ C \in \mathbb{R}^{m \times p}, ~~~ N=N^{\text{T}} > 0 \in \mathbb{R}^{m \times m} $$ are known, and $$ M=M^{\text{...
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Wrong dimension for linear matrix inequalities in control

Given a linear system with A matrix 4x4 and B matrix having 2 columns 4 rows I want to solve a LMI problem to determine the K that stabilize the linear system. \begin{equation} \dot{x} = Ax + Bu\\ u = ...
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LMI robustness to small perturbations

Let $P\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, i.e., $P=P^\top$ and $\langle Px,x\rangle\geq0$ for all $x\in\mathbb{R}^n$. Assume that, for a certain $A\in\mathbb{R}^{n\times n}$ ...
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Matrix inequality with different dimension and square root

I am trying to prove the matrix inequality which came from the Gelbrich distance. The inequality seems to be correct as I substituted some random values, but not 100% sure with that. The inequality is ...
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From where does positive semidefinite thing came

From Boyd & Vandenberghe's Convex Optimization: As I understand, all $x_i$ are scalar. So, when it says $A(x) \preceq B$, it should mean elementwise inequality. Thus for $B - A(x)$ matrix, all ...
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Why are these linear matrix inequalities equivalent? [duplicate]

Given a square matrix $A$, we want to find the symmetric matrix $P$ such that $$ A^TP + PA < 0, P > 0 \tag{1} $$ In a paper that I'm reading, the authors write that the strict linear matrix ...
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how can I find D in a way that $max_D{e^T V(D)^T S V(D) e}$? And how can I find differentiation of a matrix with respect to a vector?

I am trying to maximize $max_D{e^T V(D)^T S V(D) e}$ with respect to D and find out D. e is a (3n+1)1 vector. V(D) is (3n+1)(3n+1) matrix like: V(D) = [ I & 0 & 0 & 0 \ 0 & D & 0 &...
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generalized matrix inequality for complex Hermitian matrices

Assume having a symmetric real matrix $A$ and a skew-symmetric matrix $\Delta = [0 1; -1, 0 ]$, such that the following generalized matrix inequality holds in the PSD sense: $$\pm \frac{i}{2} \Delta\...
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How to add Linear Matrix Inequality (LMI) constraints to a Semidefinite program (SDP) in standard form

Given an SDP problem with $m$ equality constraints and one Linear Matrix Inequality (LMI) in standard form: $$ \begin{align} \min \quad & \mathbf{F}_0 \bullet \mathbf{Y} \\ \text{s.t.} \...
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How to solve an LMI feasibility problem with such an equality constraints?

My friend is now working on a simulation problem in control theory with MATLAB, where a linear matrix inequality (LMI) feasiblity problem needs to be solved at first. However, in this LMI problem, ...
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Use of the inequality $A \succeq B^{\dagger}$ when B is not invertible

In semi-definite programming (SDP), you might have an optimization problem where $A \succeq B^{-1}⪰0$ is a constraint, which implies that $A_{ii} \geq (B^{-1})_{ii}$ for all $i$. In some cases, $B$ ...
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Is the S-procedure or S-lemma valid for nonquadratic inequalities?

The s-procedure or s-lemma tries to solve a system of quadratic inequalities via a linear matrix inequality (LMI) relaxation. There are many different enunciations and if you are not sure what i mean ...
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Dualization of matrix inequalities

I am studying Linear Matrix Inequalities (LMI) in Control Theory with this lecture notes: https://www.imng.uni-stuttgart.de/mst/files/LectureNotes.pdf I am at the Dualization Section (4.4.1, page 106) ...
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Linear matrix inequalities equivalent transformation trick

I am struggling to understand a proof regarding the transformation of matrix inequalities and need your help. Thank you in advance. I am reading the following paper: https://arxiv.org/pdf/2304.03519....
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Applying an affine transformation from the matrix to vector representations of spectahedra

On page 9, Parrilo wrote$^\color{magenta}{\star}$ that the vector representation of a spectrahedron $$ S = \left\{ (x_1, \dots, x_m) \in {\Bbb R}^m : A_0 + \sum_{i=1}^m A_i x_i \succeq 0 \right\} $$ ...
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The Lyapunov inequality for a given matrix $P$

The famous Lyapunov theory says if a system matrix $A$ is stable, then the Lyapunov inequality $$A^TP+PA<0, \qquad P>0$$ is unique which depends on the negative definite matrix $-Q$, which I ...
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Transforming determining $\exists x \in \mathbb{R}^m, A(x) \succ 0$ into least squares possible?

Consider a linear operator $A: \mathbb{R}^{m} \to S^{n \times n}$, where $S^{n\times n}$ are the symmetric n by matrix. Can we turn the problem of determining if there exists $x \in \mathbb{R}^{m}$ s....
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Matrix relations involving trace and eigenvalues

Given the real symmetric positive definite matrix $A \succ 0$, consider the following inequality in $\alpha \in \mathbb{R}$. $$ A \succ \alpha I $$ where $I$ is an identity matrix of appropriate ...
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Constrained linear matrix inequality

Consider the following linear matrix inequality (LMI) $$ A - \delta B^{\text{T}} B > 0 \tag{1} \label{1} $$ where $ A \in \mathbb{R}^{n \times n} $ is a known symmetric positive definite matrix, $ ...
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convex optimization of an inequality

The motivation for this question is a relaxation of the well-known Riccati equation that will be introduced as a constraint in a convex optimization. The variable is $P\succeq0$, and the constraint is ...
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How to find the largest ellipsoid under certain constraints?

Given a symmetric positive definite matrix $\bf Q$ and a bounded set $\mathcal X$, what is the following maximum? $$ \max_{{\bf x} \in \mathcal X} {\bf x}' {\bf Q} \, {\bf x} $$ Using a Matlab program ...
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On the relationship between the Sylvester criterion and the Schur complement for positive semi-definiteness of a matrix via LMI

For some context, I see that when people try to determine the positive definiteness of a matrix, the Sylvester criterion can be of great help since the determinant is easy to calculate. For the case ...
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Proof of a matrix inequality with sorted column vectors and permutation matrices

Consider two coloum vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$. The elements in vectors $x$ and $y$ are sorted (in ascending or descending order, but the sorting order of the two vectors ...
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Can this problem be reduced to solving an LMI?

Let $A \in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{n\times m}$. We want to choose an $X \in \mathbb{R}^{m\times n}$ such that the following matrix $$ M(X) := \begin{bmatrix} A & - B X A\\ ...
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How can I express a linear matrix inequality in an expanded form?

In the paper Kalman filtering with intermittent observations by Sinopoli et al., I found the following linear matrix inequality (LMI) $$ \begin{bmatrix}X - (1-\lambda)AXA^T & \sqrt{\lambda}F \\ \...
mhdadk's user avatar
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Outer approximation of union of ellipsoids via $\log\det$ objective and LMI constraint

Given $A_i$, $b_i$, $c_i$ for $i=1,2,..p$ ($p$ is known), where $A_i\in \mathbb{R^{3\times3}}$ are symmetric positive definite matrices, $b_i\in \mathbb{R^3}$ and $c_i\in \mathbb{R}$. Let the ...
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Upper bound of a matrix using trace

I was reading a proof that uses the following matrix relation; $$ A \leq \text{Tr}(A)I $$ Where $\text{Tr}(\cdot)$ denotes the trace operator and $I$ is the identity matrix of appropriate dimension. ...
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Bound on (matrix spectral norm) $\|A-B\|$ from bound on $\|e^A-e^B\|$

Let $A,B$ be orthogonal matrices satisfying $\|A\|, \|B\| \geq c$. Now provided with $ \| e^A-e^B\|\leq \varepsilon e^{c} $, how do I show that $\| A-B \| \leq \varepsilon$? My intuition comes from ...
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Schur complement

I am trying to understand what steps need to be done to go from $P-A^TPA\succ0$ (with $P \succ 0$ and $G$ can be any matrix) to $$\begin{bmatrix} P & A^TG^T \\ GA& G + G^T - P \end{bmatrix} \...
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Equivalent Positive definite matrix transformation

Assume that we have matrix \begin{equation} \begin{bmatrix} X_1 & X_2 & X_3 \\ X_2^\top & X_4 & X_5 \\ X_3^\top &X_5^\top & X_6 \end{bmatrix} \succ 0 \end{equation}, where $X_i$...
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Is this matrix identity correct?

Is the following matrix identity correct $$L^TS+S^TL \preceq \alpha^{2} L^TL+\alpha^{-2}S^TS$$ I don't exactly know the interval where $\alpha$ lies! Would be great if someone pinpoint it.
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Lyapunov stability for a nonlinear system including two subsystems, Linear and nonlinear equations

I am studying Lyapunov stability for a nonlinear systems as follow: \begin{equation}\label{nonlinear_stab_changeCord} \begin{cases} \begin{split} L\dot{\widetilde{x_{1}}} &= c_1 \widetilde{...
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Reformulation of matrix trace maximization to an SDP

Consider the following optimization problem: \begin{equation*} \begin{split} \max_{X,Y} \;& \mathrm{Tr}[A Y] + \mathrm{Tr}[(B X B^\top)^{1/2}]\\ \mbox{s.t.} \; & Y = C - F^\top B^\top (X + D)^{...
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How do I go about proving the feasibility of this LMI?

I have the given Linear matrix inequality $$ {X^T}{\left( {{P^{ - 1}} - {\gamma ^{ - 2}}}I \right)^{ - 1}}X \succeq {\left( {X - C} \right)^T}{\left( {{T^{ - 1}} - {\gamma ^{ - 2}}}I \right)^{ - 1}}\...
SAM's user avatar
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Linear matrix inequality for nonlinear system

Considering the nonlinear control system: $$ \dot{x}=Ax(t)+B\phi(y) $$ where $$ \phi(.): \mathbb{R} \mapsto \mathbb{R} $$ is a scalar sector-bounded nonlinearity, viz $$ \phi \in Sector[\alpha,\beta] $...
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Condition on singular values of two matrices

For symmetric matrices $M$ and $N$, it is easy to verify that $\sigma_{max}(M)\leq\sigma_{min}(N)$ is a sufficient condition for $M\leq N$, where $\sigma_{max}(.)$ and $\sigma_{min}(.)$ denote the ...
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LMI reformulation

In Data-driven stabilization of discrete-time control-affine nonlinear systems: a Koopman operator approach, I read that the following LMI $$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P &...
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How do you convert the following optimization problem with LMI constraints from the standard form given to the Semi Definite Programming(SDP) form?

We are currently working on an engineering project involving convex optimization. The project relates to a low earth orbit spacecraft rendezvous. In the course of this project, we are required to ...
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$A$, $B$, and $A-B$ are non negative definite matrices. How to show that $\det(A) \geq \det(B)$?

Suppose $A,B\in \mathbb{R}^{n \times n}$ are non negative definite matrices. We have already know that $A-B$ is also non negative definite. How to show that $\det(A) \geq \det(B)$, if $\det(A)$ means ...
Zifeng Zhang's user avatar
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Schur complement and positive semidefinite cones

I know that my question might be trivial but I would appreciate your feedback. I know that the Schur complement can be used to express a quadratic inequality as a positive semidefinite matrix and vice-...
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Show that $\{x \mid A_0 + \sum x_i A_i \succcurlyeq 0 \}$ is convex

Let $A_0, A_1,\dots,A_m$ be symmetric matrices. Let $x \in \mathbb R^m$ and define $$A(x) := A_0 + \sum_{i=1}^m x_i A_i$$ Show that the set $C := \{x \mid A(x) \text{ is positive semidefinite} \}$ ...
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