# Questions tagged [lmis]

Linear Matrix Inequalities (LMIs)

73 questions
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### Can this Matrix Inequality be convexified/linearized?

(see Picture) I have the following matrix inequality in \Pi, U, V=V^T, and W=W^T, where size(V)>size(W). I would like to know whether this MI can be linearized. Any suggestions are welcome... If any ...
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### Describing Constraints Using Linear Algebra (Convex Optimization)

I've been learning Convex Optimization but one thing that really confused me in class was how exactly to recast a given set of constraints in matrix form, so that it can be solved using CVX. For ...
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### How to write this inequality in terms of Schur Complement?

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $\lambda$ such that the the matrix is definite positive, I got a little bit confused ...
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### 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 ...
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### Transforming matrix inequality into LMI

Is there a variable change I could do in order to transform the matrix inequality below $$A^T P A - P + Q + K^T B^T P A + A^T P B K + K^T(B^T P B + R) K \leq 0$$ into an LMI? My variables are ...
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### How is this a linear matrix inequality?

In example 3.4 of Stephen Boyd & Lieven Vandenberghe's Convex Optimization, it is mentioned that the last condition of $$\text{epi}=\{(x,Y,t) \mid Y\succ 0, x^TY^{-1}x\leq t\}$$ is a linear ...
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### Show LMI $F(x)\succ0$ is feasible if and only if the LMI $F(x) \succeq I_{n \times n}$ is feasible

Let $F : V \to \Bbb S^{n\times n}$ be a linear map, where $V$ is a vector space and and $S^{n\times n}$ is the set of $n \times n$ symmetric matrices. Prove that the LMI $F(x) \succ 0$ is feasible if ...
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### Writing an accurate SDP solver in Matlab

As part of a research project I'm supposed to write an semidefinite programming solver in Matlab (similar to SDTP3, MOSEK, SEDUM, etc) except it needs to be able to solve to many significant digits ...
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### SOSTools - Getting example to work?

EDIT: Changed example to a more suitable one for discussion (smaller): I am trying to solve example 4.2 from this paper: Analysis of Non-polynomial Systems using the Sum of Squares Decomposition. ...
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### How to transform the following inequality into LMI (linear matrix inequality)?

Let $B$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$, $Z$ and $T$ by solving the following inequality \begin{...
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### How can we solve this matrix inequality?

Let $A$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$, $Y$, $Z$ and $T$ by solving the following inequality \...
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### How can we solve the following matrix inequality?

Let $A$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$ and $Y$ by solving the following inequality \begin{...
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### Infinity norm of matrix as inequality constraint in optimization

Imagine the optimization problem below, where the matrix $\ell_\infty$ norm appears as an inequality constraint: \min_{X} \| XB - C\|_F^2 \\ \text{s.t. } \|AXB - B\|_\infty \leqslant ...
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### LMI result for quadratic stability of norm-bounded differential inclusions

I am trying to understand how equation (5.14) is derived in the textbook Linear Matrix Inequalities in System and Control Theory by Boyd et al. Specifically, the result deals with the quadratic ...
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### Algebraic Riccati Inequality Solution via LMI

I'm facing the following problem obtaining the solution of the Discrete Algebraic Riccati Inequality. Notation and assumption: $\succeq, \succ,\preceq,\prec$ refers to matrix definiteness ...
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### Expressing (epigraph) inequality involving trace of matrix product as a matrix inequality

I want to express an inequality of the form $$\mbox{tr} (A^{-1}B)\leq t$$ as a matrix inequality, where $A$ is positive definite and $B$ is positive semidefinite. In particular, the matrix ...
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### How to solve this matrix inequality?

Let $C$ be a given (known) matrix and let $\theta$ be a given (known) positive real. We would like to determine the matrices $X$ and $Y$ and diagonal matrix $P$ solving the following inequality \...