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3

Extending Robs answer slightly, taking into account that you asked about ILP (which I interpret as mixed-integer linear program), the constraint is MILP-representable as long as $f$ is MILP-representable (thus allowing you to have piecewise affine functions such as min/max/abs/general pwa etc)


10

Your constraint is equivalent to $$x_i \le f(x_j) \quad \text{for $j<i$},$$ so it is linear if $f$ is linear.


4

Try adding valid constraints $$ y_{i,j} \le \sum_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,k} \quad\text{for $(i,j)$ such that $\hat{y}_i = 0$ and $j \in \tilde{V}\setminus\{i\}$} $$ that enforce the logical implications $$(y_{i,j} \land \lnot\hat{y}_i \land [j \in \tilde{V} \setminus\{i\}]) \implies \bigvee_{(i,k): k \in \tilde{V} \setminus \{j\}} y_{i,...


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

A short look into the literature shows that mixed integer multi objective algorithms are still in an early stage with people proposing different formulations and approaches. I am not aware of any open solvers that could be readily used even outside the R ecosystem. So i see multiple options for you: Weighted sum approach and use the package GA as prubin ...


0

Your question title mentions "evolutionary algorithm", so (a) I assume you are comfortable with a heuristic solution and (b) I assume you have some familiarity with evolutionary algorithms. There is an excellent genetic algorithm package for R, named GA. I think there may be a few other genetic algorithm packages for R, but this is the only one ...


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