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It's not clear to me from the problem description what's being optimized, so I'm going to try to rephrase the problem, then answer the rephrased version. Also, the presence of two streams ($X$ and $Y …

Suppose you want to rotate $k$ vectors at once. Consider the matrix $M$ formed by concatenating these $k$ vectors (so $M$ is an $n\times k$ matrix, and each column of $M$ is one of the original vecto …

Another way to solve the problem might be to reduce it to a traditional version of the TSP and then use a free, off-the-shelf solver like one of these. At the very least, it should be a relatively ea …

[Edit: I had originally posted a proof that finding the minimum value was NP-hard; in the comments below, Brendan McKay pointed out how to convert that into a proof that finding the maximum value is N …

Let Property A be the property described above. Let Property B be the property that $f_2=g \circ f_1$ for some strictly monotonically increasing function $g:R\rightarrow R$. Then Property A and Pro …

This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define $$ \mathbf{K}(\phi)_{i,j} …

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\operator …