All Questions
8 questions
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53
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Relations between the optimal solutions of two related SDPs
In system theory, we often encounter Semi-Definite Programs (SDPs) with Linear Matrix Inequality (LMI) constraints, such as those presented in this paper. I have introduced new variables based on the ...
- 4,484
1
vote
2
answers
69
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Monotonicity of kernel matrices with respect to hyperparameters
Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
- 801
0
votes
0
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159
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Double summation of matrices as constraints in convex optimization in CVX
I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53:
\begin{align}
\text{minimize} &\qquad s\\
\text{subject to} & \...
1
vote
0
answers
78
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Is there an efficient way to do semidefinite programming with a Lyapunov equation constraint?
I am trying to numerically solve semidefinite programs of the form
$$\begin{array}{ll} \underset{X,Y}{\text{minimize}} & \operatorname{tr}(AX)\\ \text{subject to} & BY + YB = X\\ & X, Y \...
- 31
0
votes
2
answers
1k
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Matrix norm minimization and matrix inner product
One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, Convex Optimization, p. 170).
Consider:
\begin{equation}\label{eq:Lasse}
\begin{aligned}
&\min_{\mathbf{x}}
& &...
- 345
4
votes
1
answer
254
views
Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices
For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices:
$$\Pi(A)=\mathrm{argmin}_{M\...
- 41
12
votes
2
answers
800
views
A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
- 2,712
4
votes
1
answer
286
views
Explicit formula for an LMI solution
Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that $X-...
- 6,962