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Problem with implementing squared terms in the objective function

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:

enter image description here

$x$: decision variable

$1$: column of ones

$k$: squared matrix

2- Summation notation:

enter image description here

$x$: 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!

One Answer

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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