Questions tagged [linear-matrix-inequality]

Linear Matrix Inequalities (LMIs)

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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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Frobenius norm inequality for product of matrices with kronecker product structure

Consider three integers $p,n,t$. Consider a matrix $M\in R^{pt\times pt}$ symmetric positive definite with eigenvalues at most one of size $(pt\times pt)$, a matrix $A\in R^{t\times t}$ and a matrix $...
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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 ...
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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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Bound on Matrix Norm From Bound on Vector Norm

This problem has come up in attempting to apply the results of [1] to the original Kalman filter. Suppose we have a $d \times d$ matrix $F$ and a $q \times d$ matrix $H$ satisfying $$\left\vert H\,F\,...
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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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Does the "Riccati LMI" implies boundedness?

Assume a matrix $P\succeq0$ satisfies the "Riccati LMI" \begin{align} \begin{pmatrix} FPF^T - P + M & FPH^T + S\\ HPF^T+S^T& HPH^T + I \end{pmatrix}\succeq0 \end{align} with $M\succ0$...
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Trace minimization with additional constraint on $X$

Consider the following SDP problem: \begin{align*} \min_X \; & \mathrm{Tr}[AX]\\ \mathrm{s.t.}\; & X \succeq 0\\ & X \succeq \begin{bmatrix}0 & 0.5\\0.5 & 0\end{bmatrix}, \end{...
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What is the meaning of the ≼ Symbol in the Context of Matrix Inequality with Symmetric Matrices

I saw the symbol ≼ in a textbook, and I am not quite sure what it means. The textbook says it represents matrix inequality, but again, I do not understand what that means. My best guesses are that the ...
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How can you convert the following form of the Riccati equation into an LMI?

On page 3 of the EE363 notes on linear matrix inequalities, the following Riccati inequality is listed $$ 0 \leq A^TPA+Q-P-A^TPB(R+B^PB)^{-1}B^TPA, \qquad P \geq 0 $$ I know that the Schur's ...
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Common factor in sum of transpose matrices

I have the following linear matrix inequality: $(B K_1)^T + BK_1 < -2A$ where B is 2x1, $K_1$ is 1x2 and A is 2x2. Is it possible to find $K_1$ as a common factor in the left-hand side of the ...
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H2-filtering of unstable LTI systems. Can this problem be reformulated as convex optimization problem with LMI constraints?

Consider the discrete-time generalized LTI plant with minimal state-space realization $$x_{k+1}=A_d x_k + B_{d1}w_k\\z_k=C_{d1}x_k+D_{d11}w_k\\y_k=C_{d2}x_k+D_{d21}w_k$$ For the Schur-stable $A_d$ ...
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Necessary and sufficient condition for a matrix to be positive definite

Consider a blocked real symmetric matrix $A\in\mathbb{R}^{2n\times 2n}$. Given $M\in\mathbb{R}^{n\times n}$, consider the matrix production $$B=\begin{pmatrix}I_n\\M\end{pmatrix}^TA\begin{pmatrix}I_n\\...
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How to prove that a certain block matrix is positive semi definite, which depends on a undetermined submatrix

How should I proof the following matrix $$M = \begin{pmatrix} Z-A^TZA & -A^TZB\\ -B^TZA & -B^TZB \end{pmatrix},$$ to be positive semidefinite? The matrices $A\in \mathbb{R}^{n\times n}$ and $ ...
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Can I reformulate the given SDP such that the main constraint becomes and LMI?

I am new to SDP and LMI's and trying to solve an optimization problem of the following form: \begin{equation} \begin{aligned} \text{maximize} \quad & \sum_{j=1}^k w_j\\ \text{subject to} \quad &...
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Linear matrix inequality and convex epigraph

In example 3.4 of Boyd & Vandenberghe's Convex Optimization, function $f : \mathbb{R}^n \times \mathbb{S}^n \to \mathbb{R}$, defined as $$f(x, Y) := x^T Y^{-1}x$$ is convex on $\mathrm{dom} f = \...
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Specify LMI terms in MATLAB when lhs and rhs have different block partitions

I'm trying to specify a LMI term via lmiterm function in MATLAB. The left-hand side (lhs) and the right-hand side (rhs) in my question have different block structures. For example, consider $$M^T\text{...
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LMI-Solution Invariant to the Initial Conditions

It is known from R. Bellman that the value of the functional $J = \int_{0}^{\infty}xWx \ dt, \ W>0$ along the solution of the linear time-invariant system $\dot x = Ax, \ x(0)=x_0$, with Schur ...
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An LMI transformation problem

Given matrices $Q>0,F,A$ and a number $\alpha\in(0,1)$, find some $P>0,X,\Psi$ such that $$ \begin{aligned} \Psi^T P \Psi\leq \alpha P\\ \begin{bmatrix} A^TP+PA-...
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Find $(P,\Psi)$ such that $\alpha P\geqslant \Psi^TP\Psi\geqslant \beta P$

Given constants $\alpha\geq \beta>0$, find a nonsingular matrix $\Psi\in\mathbb{R}^{n\times n}$ and a postive definite matrix $P$ such that $$ \begin{aligned} \alpha P\geqslant \Psi^TP\...
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How to check the feasibility of standard LMI using Matlab/CVX?

In the wikipedia page of LMI, the standard form is given by $$A_0+y_1A_1+y_2A_2+\cdots+y_mA_m \succeq 0,$$ where $A_i$ are $m\times m$ symmetric matrices and $y_i$ are real vectors, $i=1,2,\ldots m.$ ...
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Why is the boundary of spectrahedra “more pointy” at matrices of lower rank?

In the following expository article about spectrahedra, it is established informally that the boundary of spectrahedra is “more pointy” at matrices of lower rank. Cynthia Vinzant, What is a... ...
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If $0\leq X \leq \text{Id}$ and $0\leq A$, then $XAX \leq A$?

Let $X,A\in\mathbb{C}^{n\times n}$ and suppose $$ 0 \leq X \leq \text{Id}, \quad 0\leq A, $$ where $\text{Id}\in\mathbb{C}^{n\times n}$ denotes the identity matrix. Is it true that $$ XAX \leq A, $$ ...
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Schur complement for linear matrix inequality (LMI)

Given the following inequality \begin{align} & \gamma \left( Q - (A Q + BY)^T Q^{-1} (A Q + BY) \right) - Y^T R^{1 \over 2} R^{1\over 2} Y - Q Q_1^{1\over2} Q_1^{1\over2} Q \succeq 0 \tag{1} \end{...
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On two different spectral radius bounds

Let $\rho(A)$ denote the spectral radius, i.e. $\max_i |\lambda_i|$ with $\lambda_i$ being the eigenvalues of $A$. The bound $(\rho(A))^k\leq ||A^k||_2$ leads to: $$ \rho(A)\leq \tau^\frac{1}{k}\quad\...
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Help on Solving Frobenius Matrix Norm Inequality

Problem:$\alpha\leq \left \| (\lambda A+B)^{-1} \right \|_{F}\leq \beta$, where $\alpha$, $\beta$ and $\lambda$ are postive constants, $A$ and $B$ are matrix; $A$ and $\lambda A+B$ are invertible but $...
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Formulate a large LMI

I have read that this inequality $P-\sum_{j=0}^{m}\left(A^{(j)}+B^{(j)} K\right)^{T} P\left(A^{(j)}+B^{(j)} K\right) \succ 0$ can be reformulated into a large LMI $\left[\begin{array}{ccccc}S & \...
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Can matrix inequality induces matrix norm inequality for positive semidefinite matrix?

I am wondering whether matrix inequality induces matrix norm inequality for positive semidefinite matrix. For example, considering two positive semidefinite matrices $A$ and $B$, when $A \succeq B$, ...
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Condition for equality in the triangle inequality for matrix norms

Say I have two matrices $A$ and $B$, which necessarily satisfy the triangle inequality for matrix norms \begin{equation} ||A+B||\leq||A||+||B||\;. \end{equation} For a general matrix norm is it ...
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Bounding norm of difference of inverse of two PSD matrices

I am going through this paper (https://arxiv.org/pdf/1902.07826.pdf) and I am stuck at the proof of Lemma 7 (on page 22). The goal is to prove that for any two PSD matrices $M, N$ of same dimensions, ...
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How to solve Linear Matrix Inequility to get suitable quadratic Lyapunov function for a linear dynamical system with non-hyperbolic equilibrium point?

I am trying to establish stability of linear dynamical system ($\dot{x}=Ax$) with non-hyperbolic equilibrium point. While soling linear matrix inequality ($A^\top P+PA \prec 0$) based on a suitable ...
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Absoute sum of diagonal elements no more than absolute sum of eigenvalues

Suppose $A$ is an $n\times n$ matrix, and $\lambda_1, \lambda_2, \ldots, \lambda_n$ are its eigenvalues. Prove that $$ \sum_{i = 1}^n \lvert{A_{ii}} \rvert \leq \sum_{i = 1}^n \lvert \lambda_i\rvert. ...
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Matrix inequality of inverses of sum of two matrices

$A$ and $B$ are two real symmetric non-singular matrices but not necessarily sign definite. Suppose $A+B$ is non-singular. I want to find a $\lambda \in \Bbb R$ such that $$(A+B)^{-1}\leq(A+\lambda I)^...
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Is it always possible to decide sign of real part of an eigenvalue of a matrix by solving linear matrix inequality?

To establish the sign of the real part of the eigenvalues of a real square matrix $A$, we usually try to find a symmetric positive definite matrix $P$ verifying the matrix inequality $A^\top P + PA \...
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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 $$ ...
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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 ...
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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 ...
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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} ...
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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 +...
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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 & ...
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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?
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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 ...
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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 ...
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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 ...
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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^{...
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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 \...
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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 ...
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Upper bounding the largest singular value of a matrix $X$ via LMI — 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: ...
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2 answers
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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: $\...
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