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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