# Questions tagged [semidefinite-programming]

Semidefinite programming can be regarded as an extension of linear programming. In a semidefinite program, the goal is to optimize a linear function over the intersection of the cone of positive semidefinite matrices with some affine space.

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### Perturbation of positive semidefinite matrix

Consider an $n\times n$ matrix $A$ that is positive semidefinite and has rank $n-1$, so there exists exactly one eigenvector $v$ such that $Av=0$. Let now $B$ be a symmetric matrix such that $v^TBv=0$....
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### Positive-definite block matrix with constant block sums

Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm \times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the ...
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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} & \...
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### A variant of the elliptope relaxation

Given a p.s.d. matrix $A$, one may want to find: $$\max_x x^t A x \mbox{ such that } x \mbox{ has entries }1 \mbox{ or } {-1}.$$ This hard problem has a well known relaxation based on the so called ...
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### When does a finite metric induce a matrix norm?

If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit ...
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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}} & &...
### Matrix inequality $a X \succeq arcsin(X)$ for some $a > 0$
Let $X \in S^{n}_{+}$ be a positive semi-definite matrix with $X_{ii} = 1$ for all $i \leq n$ (thus $X$ is a correlation matrix). Since $X$ is positive semi-definite, we have $|X_{ij}| \leq 1$ for any ...