Operations Research Asked by PoofyBridge on August 19, 2021
I’m trying to implement either one of these objective functions, but I’m having a hard time with the squared terms. I’m attaching both so you can take a look at the structure and see if you can give me any tips. Is there any way to implement either one of them?
1- Matrix notation:
$x$: decision variable
$1$: column of ones
$k$: squared matrix
2- Summation notation:
$x$: decision variable
$m$: degree of the node i
$rho$: parameter that takes into account the influence of the neighbors that surround node i
$a$: terms of the adjacency matrix. Shows if nodes i and j are connected
Thank you in advance!
It's relatively easy to write $(1^{T}Kx)^{2}$ in standard quadratic form.
Since $1^{T}Kx$ is a scalar,
$(1^{T}Kx)^{2}=(1^{T}Kx)(1^{T}Kx)^{T}=1^{T}Kxx^{T}K^{T}1$.
Using the cyclic property of the trace of a product of matrices,
$1^{T}Kxx^{T}K^{T}1=mbox{tr}(1^{T}Kxx^{T}K^{T}1)=mbox{tr}(x^{T}K^{T}11^{T}Kx)=x^{T}(K^{T}11^{T}K)x$.
Unfortunately, $K^{T}11^{T}K$ will be dense, so if $x$ is large you'll probably run out of storage.
Answered by Brian Borchers on August 19, 2021
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