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For continuous LP, problems with millions of nonzeros are solved routinely. I'd expect problems with 10 millions nonzeros to be solvable, barring numerical issues. You can find some benchmarks here, but they do not include the best commercial solvers, which I remember solve all problems. This is my guess from experience, and I think common knowledge in the ...


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Let $a_{ij}$ represent member of matrix $A$. This can either be the constant $0$ if you want $a_{ij}$ to be $0$ in the solution matrix, or a variable for a linear programming problem. Then you want to add linear constraints: \begin{alignat}{2} &\!\min_{a} &\qquad& \sum_{i} s_i\\ &\text{subject to} & & \forall j \left(\sum_{...


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Your condition is effectively excluding $(0,1,1)$ and $(1,0,1)$ corners of the unit cube. Thinking of it pictorially gives you a quick formulation: (i) construct the cube, (ii) chop those corners off and (iii) ensure integrality. In (ii), the corners we want to get rid of are on the $BC$ and $AC$ planes. We can cut from the correctly aligned diagonals on ...


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You can easily impose the constraint that a given city should be visited before a given location in the sequence. I.e., that city 5 should be one of the three first visited: node_order[5 - 1] <= 3 Or that city 7 should be visited as the third, fourth, or fifth: node_order[7 - 1] <= 5, node_order[7 - 1] >= 3 This does not exactly impose the ...


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