edschaal
commented
1 month ago Dear Sebastian,
Sorry for the late reply. I have been thinking about what you sent me and I must say that I just cannot comprehend what I meant with this footnote. I am actually thinking what I was describing was for something different (this comment on the inversion of the linear system is something you can use in the model without congestion...). I just can no longer make sense of it and we should correct this appendix. This sometimes happens with footnotes that lie around the text and that we never check.
This being said, we wrote this 5 years ago, so I might be missing something today. But I think it should still be possible to solve this case with duality. Let me try to work out a quick note on that and I'll get back to you.
Thanks,
E.
On Sat, Oct 5, 2024 at 11:03 AM Sebastian Krantz @.***> wrote:
Hi @edschaal https://github.com/edschaal. I am trying to implement the dual model with cross-good congestion, because I think this is necessary to do realistic simulations on very large networks. I've started doing this in Julia, but want to implement it in MATLAB as well if I get it to work properly. You can see my inaugural Julia implementation here https://github.com/OptimalTransportNetworks/OptimalTransportNetworks.jl/blob/development/src/models/model_fixed_duality_cgc.jl. One aspekt I don't understand is how to evaluate the Lagrangian with only aggregate flows $Q{jk}$ and invert the linear system defining $Q{jk}^n$ after the optimization - as per Footnote 48 in Section B1 of the Supplementary Material https://raw.githubusercontent.com/OptimalTransportNetworks/OptimalTransportNetworkToolbox/main/docs/paper_materials/OTN_supp_material.pdf. In my reading the flow constraint is $D_j^n + \sumk Q{jk}^n \leq Y_j^n
- \sumk Q{kj}^n$ and the Lagrangian is $\sum_j \omega_j L_j u_j - \sum_j \sum_n P_j^n (D_j^n + \sumk Q{jk}^n - Y_j^n - \sumk Q{kj}^n)$, so I don't understand how this can be evaluated without $Q{jk}^n$. I my preliminary Julia implementation I have defined $Q{jk}^n$ as an additional $ndeg \times N$ optimization variable alongside a constraint that ensures that $Q_{jk} = \left(\sumn m^n (Q{jk}^n)^\nu\right)^{1/\nu}$. This of course limits the value of the dual approach. Your input on this would be much appreciated.
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Hi @edschaal. I am trying to implement the dual model with cross-good congestion, because I think this is necessary to do realistic simulations on very large networks. I've started doing this in Julia, but want to implement it in MATLAB as well if I get it to work properly. You can see my inaugural Julia implementation here. One aspect I don't understand is how to evaluate the Lagrangian with only aggregate flows $Q{jk}$ and invert the linear system defining $Q{jk}^n$ after the optimization - as per Footnote 48 in Section B1 of the Supplementary Material. In my reading the flow constraint is $D_j^n + \sumk Q{jk}^n \leq Y_j^n + \sumk Q{kj}^n$ and the Lagrangian is $\sum_j \omega_j L_j u_j - \sum_j \sum_n P_j^n (D_j^n + \sumk Q{jk}^n - Y_j^n - \sumk Q{kj}^n)$, so I don't understand how this can be evaluated without $Q{jk}^n$. I my preliminary Julia implementation I have defined $Q{jk}^n$ as an additional $ndeg \times N$ optimization variable alongside a constraint that ensures that $Q_{jk} = \left(\sumn m^n (Q{jk}^n)^\nu\right)^{1/\nu}$. This of course limits the value of the dual approach. Your input on this would be much appreciated.