For for instance, we need to figure a \(w\) such that:
\begin{equation} f = w^{\top}\mqty[f_1 \\ \dots\\f_{N}] \end{equation}
where weight \(w \in \triangle_{N}\).
To do this, we essentially infer the weighting scheme by asking “do you like system \(a\) or system \(b\)”.
- first, we collect a series of design variables \((a_1, a_2, a_3 …)\) and \((b_1, b_2, b_3…)\) and we ask “which one do you like better”
- say our user WLOG chose \(b\) over \(a\)
- so we want to design a \(w\) such that \(w^{\top} a < w^{\top} b\)
- meaning, we solve for a \(w\) such that…
\begin{align} \min_{w}&\ \sum_{i=1}^{n} (a_{i}-b_{i})w^{\top} \\ \text{such that}&\ \bold{1}^{\top} w = 1 \\ &\ w \geq 0 \end{align}
unlike the rest of everything, we are MAXIMIZING here idk why
example
assume: if we prefer \(a\) to \(b\), then \(w^{T} a > w^{T} b\).
Let’s say we had two bags, each with \(a = \qty(1,3,6)\) and \(b = \qty(7,1,2)\).
This means:
\begin{equation} 1 w_1 + 3 w_2 + 6 w_3 > 7 w_1 + w_2 + 2 w_3 \end{equation}
Doing algebra, this gives:
\begin{equation} -6 w_1 + 2 w_2 + 4 w_3 > 0 \end{equation}
Recall this is also a probability, so we have:
\begin{equation} 1 - w_1 - w_2 = w_3 \end{equation}
Finally, solving this gives:
\begin{equation} 5w_1 + w_2 < 2 \end{equation}
This is what we call a halfspace, which further bounds weights that are possible. Its a line which bounds the space of weights down. Combining each of the halfspaces together gets a piecewise linear graph. Taking say the centroid of the remaining space will give you the desired result.