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I have been working on a problem for a couple months now and I figured I might be able to find the answer if I could minimize the set of players preferences. I am trying to minimize a space where there is $n$ players with $m$ policies and a discrete set of possible coalitions. Each player not only has preferences within a policy domain $(0\leq X_i\leq 1_j$ from $i=1,..,n$ and $j =1,...m)$ but also between them $(W_{ij}\geq1)$. and they are trying to minimize the suggested policy space $(Y_j)$ to their preffered policy set $(X_{ij})$ while incentivizing a subset of players to join their coalition.

given $W_{ij}$ and $X_{ij}$ I am trying to find the policy set $(Y_j)$ that minimizes the equation below with the constraint that for each $j,$ $0\leq Y_j\leq 1$.

The problem for me is I am not sure how to deal with the absolute value in a minimization problem? How would I go about finding the vector $Y_j$ given $W_{ij}$ and $X_{ij}$?

$$\min\sum_{i}^n\sum_{j}^m \lvert(W_{ij}(X_{ij}-Y_j)\vert$$

Could add an arbitrary variable $T_{i}$ and take each $$W_{ij}(X_{ij}-Y_{j})>=-T_{i}$$ $$W_{ij}(X_{ij}-Y_{j})<=T_{i}$$ then add slack variables and re-arange; adding the constraint $$Y_{j}<=1$$ and the objective function $$\sum_{i}^nT_{i} $$ and solve it using Simplex method linear programming?

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You are almost correct. But rather than $T_i$, you should consider $T_{ij}$.

Let $T_{ij}= \left| W_{ij}(X_{ij}-Y_j)\right|=max(W_{ij}(X_{ij}-Y_j),-W_{ij}(X_{ij}-Y_j))$

$$T_{ij}\geq W_{ij}(X_{ij}-Y_j)$$

$$T_{ij}\geq -W_{ij}(X_{ij}-Y_j)$$

and minimize $\sum_{i=1}^n \sum_{j=1}^m T_{ij}$

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