Sudoku mathematically, MILP? My homework contains a word (freely-translated) "target-function" that I should generate somehow for 9x9 sudoku solver with some MILP problem. But I am bit lost what they mean. I have sofar described the problem as below. Is this the target function for MILP problem (apparently meaning mixed integer linear problem or something like that)?
\begin{cases}
\forall r\in {1,2,...,9}\sum_{c=1}^{9}V_{rc}=45 \\
\forall c\in{1,2,...,9}\sum_{r=1}^{9}V_{rc}=45 \\
\forall r\in{1,2,...,9}\forall n\in {1,2,...,9} \sum_{c=1}^{9}x_{rcn} = 1 \\
\forall c\in{1,2,...,9}\forall n\in{1,2,...,9}\sum_{r=1}^{9}x_{rcn} = 1 \
\end{cases}
where $x_{rcn}$ is a binary function with $r$ for row,  $c$ for column and $n$ for number between 1 and 9. $V_{rc}$ means the number $n$ in the position $(r,c)$. When I look at wikipedia about the mathematics of sudoku here, I am worried that my way of formulating the problem is not right at all. When I google for MILP, I do not get descriptive hits -- what does this MILP function thing mean?
I have not yet understood how I can formulate the above messy formulates more elegantly, perhaps with the group tabels...sorry I am bit puzzled with the tables in the wikipedia article.
 A: The linear program can be formulated just using the binary 0/1-variables 
$x_{rcn}$ with the meaning $x_{rcn} = 1$ if and only if there is the number $n$ in the 
cell $rc$, i.e in the row $r$ and the column $c$. 
There are $9*9*9 = 729$ variables $x_{rcn}$. 
The constaints are:
(1) There is exactly one number in a cell:
$$x_{rc1} + x_{rc2} + ... + x_{rc9} = 1; r, c = 1, .., 9$$
(2) The number $n$ appears exactly once in each column and each row:
$$x_{r1n} + x_{r2n} + ... + x_{r9n} = 1; r, n = 1, .., 9$$
$$x_{1cn} + x_{2cn} + ... + x_{9cn} = 1; c, n = 1, .., 9$$ 
(3) The number $n$ appears exactly once in each of the nine 3x3 boxes:
$$x_{11n} + x_{12n} + ... + x_{33n} = 1; n = 1, .., 9$$
$$...$$
$$x_{77n} + x_{78n} + ... + x_{99n} = 1; n = 1, .., 9$$
The number of constrains is:
(1) 81,
(2) 81 + 81 = 162,
(3) 81,
totally 324.
Additional constrains are given by predefined positions. For example if
there is the number 3 in the position (2, 5), then $x_{253} = 1$.
There is no explicit target function. If a target function is needed,
then any linear function of the free binary variables can be taken, e.g. $x_{111} = min/max$. 
