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### Which solvers should I use to solve large, but extremely sparse LP problems with 100-500 thousand variables?

One of the best-performing free LP solvers exploiting sparsity is HiGHS (https://github.com/ERGO-Code/HiGHS). It allows passing a starting point. For comparing relative performance of a number of LP ...

### For an ILP relaxed to LP is the LP solution objective always less than the ILP solution?

If the integer linear programming is a minimization problem then whose linear relaxation is a lower bound for the original MILP model, and for the maximization problem it is an upper bound. In the ...

### Which solvers should I use to solve large, but extremely sparse LP problems with 100-500 thousand variables?

Based on the Mittelmann benchmarks, I would agree that HiGHS is a contender, as is Clp (which is faster than HiGHS on some test problems, slower on others). You might also want to look at SCIP, which ...

### How to handle strict inequalities?

In practice, most solvers are implemented to use fast but inexact floating-point arithmetic and thus can only ever guarantee to satisfying the optimality conditions of your problem to within some ...
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### Linearizing if else conditions in ILP

Your first constraint enforces more than was asked. When $X_{ij}=0$, it forces $\sum_k Y_{jk}=0$, hence $Y_{jk}=0$ for all $k$. To enforce only $$X_{ij}=1 \implies \sum_k Y_{jk}=1,$$ you can instead ...
1 vote
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### how do I find the dual when a variable has an upper bound?

You can imagine the upper bound for the variable is another constraint. $\max: c^Tx$ $\text{s.t.}$ $Ax = 0$ --> $π_1$ $x ≤ d$ --> $π_2$ $0 ≤ x$ Therefore, you will obtain the following dual form....
1 vote

### How to select intermediate nodes in a network?

Option 1: Run Dijkstra's algorithm first. Then you will have a grid of positive numbers that you can use for additional constraints. E.g. require a positive number to know that a path is available ...

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