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9

Here's another single-solve solution. Replace each original variable $x_n$ with a sum of two variables, $x_n=y_n + z_n$, where $y_n$ is integer-valued and $z_n\in [0,1]$. Now define $\lbrace z_1,\dots, z_n\rbrace$ to be a type 1 special ordered set (SOS1). Assuming the solver supports SOS1 constraints, you'll end up with a solution in which at least $n-1$ of ...

8

An alternative approach that requires only one solve and no modification of the model is to modify branch and bound to prune by integrality when at most one integer variable takes a fractional value (rather than the usual requirement that all integer variables are integer-valued). You would also need to disable any presolve/cut routines that assume ...

2

The question is a bit confusing, in that you say you have a MILP, mention a continuous variable but nothing about integer variables, and then ask about the dual solution (which exists for a continuous model but not for an integer model, at least not in the usual sense of "dual"). Assuming that we are talking about a linear program with the ...

2

MILP with a fixed objective function is hard as finding a feasible solution. Since this can still encode n-queens completion or SAT it is still in the same complexity class. Depending on your problem some MILP heuristic might find a solution. Constraint satisfaction programming might be fast but it could be that encoding it into sat or pseudo boolean form ...

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