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Questions tagged [linear-matrix-inequality]

Linear Matrix Inequalities (LMIs)

A linear matrix inequality (LMI) is an expression of the form

$$A_0 + y_1 A_1 + y_2 A_2 + \cdots + y_m A_m \succeq O_n$$

where $y \in \mathbb R^m$ and $A_0, A_1, \dots, A_m$ are symmetric $n \times n$ matrices. The generalized inequality $Q \succeq O_n$ means that $Q$ is a positive semidefinite matrix.

This linear matrix inequality specifies a convex constraint on $y$.

There are efficient numerical methods to determine whether an LMI is feasible (e.g., whether there exists a vector $y$ such that LMI($y) \ge 0$), or to solve a convex optimization problem with LMI constraints. Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs. Also LMIs find application in Polynomial Sum-Of-Squares. The prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dual convex cones governing this LMI.

See also: Wikipedia

202 questions
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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 elliptope is the spectrahedron defined as…
Rodrigo de Azevedo
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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 simpler problem that does not go through when…
Morad
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Is it true that $A^2\succ B^2$ implies that $A\succ B$, but not the converse?

I remember reading somewhere about the following properties of non-negative definite matrix. But I don't know how to prove it now. Let $A$ and $B$ be two non-negative definite matrices. If $A^2\succ B^2$, then it necessarily follows that $A\succ B$,…
Guanghua Shu
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Solution set of an LMI is convex

I was going through Boyd & Vandenberghe's Convex Optimization. On page 38, the authors mentioned that the solution set of a linear matrix inequality (LMI) is convex. $$ A(x) := x_1 A_1 + \dots + x_n A_n \preceq B $$ where $A_1, \dots, A_n, B \in…
Rajat
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Writing a convex quadratic program (QP) as a semidefinite program (SDP)

Given a convex quadratic program (QP) $$\begin{array}{ll} \underset{x}{\text{minimize}} & \mathrm x^\top \mathrm Q \, \mathrm x + \mathrm r^{\top} \mathrm x + s\\ \text{subject to} & \mathrm A \mathrm x \leq \mathrm b\end{array}$$ how can one write…
Rodrigo de Azevedo
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Spectral norm minimization via semidefinite programming

Given symmetric matrices $A_0, A_1, \dots, A_n \in \mathbb R^{m \times m}$, let $A(x) := A_0 + x_1 A_1 +\cdots + x_n A_n$. How to formulate the following unconstrained spectral minimization problem as a semidefinite program? $$\min_{x \in \mathbb…
Marcus118
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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 + A X A^T + Y A^T + A Y^T \right)\\ \text{subject to}…
Morad
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Minimization of Frobenius Norm and Schur Complement

There is a famous problem Optimization of Frobenius Norm and Nuclear Norm; however, this is not I want to ask (about proximal operator). Suppose I have an easy optimization problem: $$\min_Q \|Q-Q_N\|_F$$ where $\|\cdot\|_F$ is the Frobenius…
sleeve chen
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Are $A^TP+PA<0$, $P>0$ and $A^TP+PA\leq-I$, $P\geq I$ equivalent?

Consider the LMI, where $A$ is a Hurwitz matrix: $A^TP+PA<0$, $P>0$, minimize trace(P) According to Stephen Boyd's book, the inequalities are homogeneous in $P$ and hence can by replaced with the nonstrict inequalities: $A^TP+PA\leq-I$, $P\geq I$,…
Lyapunov123
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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 LMI? Explicitly, can we show something…
Morad
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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 definite, then there does exist a positive…
ryuta osawa
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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 solving an LMI-based SDP is computationally very…
Teodorism
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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 definitions and notation in this book are, so I'll…
zxmkn
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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 and only if the LMI $F(x) \succeq I_{n \times…
M.D.
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Transform SDP with LMI to SDP of standard form

For a semidefinite program (SDP) with a linear matrix inequality (LMI), we can write it as follows. $$ \begin{array}{ll} \underset {x \in {\Bbb R}^n} {\text{minimize}} & c^T x \\ \text{subject to} & G + x_1 F_1 + x_2 F_2 + \dots + x_n F_n \preceq 0…
Finley
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