Paper: PREDICTING ACCURATE LAGRANGIAN MULTIPLIERS FOR MIXED INTEGER LINEAR PROGRAMS (ICLR-2024)

Reviewer comment (Experiments):
The approach is applicable not only to MILPs but also ILPs or 0-1 LPs. A good source of more complex decompositions is the 0-1 LP formulation of the max-apriori (MAP) inference problem in graphical models (also known as discrete energy minimization or Weighted Constraint Satisfaction Problem). This problem can be decomposed to arbitrary subproblems, each of which is itself a MAP inference problem. While tree-structured subproblems provide the same bound as the continuous (LP) relaxation, non-tree subproblems (such as cycles or planar graphs) provide strictly tighter bounds. There is a large public database of instances available for such problems.

Author response:
The two problems on which we tested our method have indeed the same structure for their subproblems. We have not tested our approach on another problem but but this is work in progress. Thanks for pointing public available datasets and MAP problems on which we could try our approach.

Reviewer comment (Experiments):
It is unclear whether the method is competitive with other methods for suboptimally computing Lagrange multipliers that are not based on learning. One example is min-marginal averaging, which—though only suboptimal without smoothing—is much faster than subgradient methods, especially when subproblems are small. For MCDN and CFL, it is hypothesized that a few iterations of min-marginal averaging, warm-started by continuous relaxation, could close a large part of the gap and be faster than prediction based on deep learning.

Author response:
We have not compared our approach with non ML methods different from the bundle method. min-marginal averaging seems to be for solving ILPs but we have mixed integer linear problems.

Reviewer comment (Experiments):
Post rebuttal, the paper is still considered not strong enough, mainly due to the limited instance class in the experiments.

Author response:
The two problems on which we tested our method have indeed the same structure for their subproblems. We have not tested our approach on another problem but but this is work in progress. Thanks for pointing public available datasets and MAP problems on which we could try our approach.

Reviewer comment (Reproducibility):
A question is raised about whether the authors plan to make the code available if the paper is accepted.

Author response:
Code and instances will be available upon the acceptation of the paper.

Reviewer comment (Novelty):
The technical contribution of the paper is very limited. Most techniques are from existing literature.

Author response:
We are sorry to see that the main contribution was unclear.
This work can be summarized as a modelling work rather than a technical work: the application of deep latent probabilistic modelling to MILP bound predictions. To our knowledge, this type of modelling is novel for this type of problem. 
Of course, there are already several works that base their models on encoder-decoder architectures, but they have a different goal, usually solving the primal problem with reinforcement learning.

We believe that the topic and amount of content is in agreement with the ICLR call for paper.

Reviewer comment (Evaluation):
The numerical results are not strong enough. The proposed method does seem to be beneficial for obtaining an initial guess of the optimal dual. But it seems like the Lagrangian dual problem itself is not computationally hard (based on the results in Section 4) even on MCND-BIG-COMVAR. The optimal dual multipliers can be found easily by BM within a few minutes.

Author response:
We agree that the optimal LMs can be obtained within 12 minutes for our more complex instances by BM. Still, this is quite long and could be replaced by a neural prediction, provided the accuracy is acceptable.

For instance a simple k-NN cannot retrieve the correct LMs (on MCND-SMALL-40):

| k   | GAP            | GAP-CR             |
| --- | -------------- | ------------------ |
| 3   | 58.12 (± 9.23) | -605.74 (± 527.38) |
| 5   | 54.08 (± 8.51) | -556.23 (± 489.36) |
| 10  | 52.32 (± 8.23) | -535.15 (± 473.33) | 

and using a MLP instead of our GNN-based probabilitic encoder and decoder is definitely not as good as our proposed method (and remember that it can only be applied to problems of the same size)

| Obj       | GAP  | GAP-CR |
| --------- | ---- | ------ |
| LR (full) | 1.90 | 85.12  |
| LR (MLP)  | 7.63 | 37.17  |

Reviewer comment (Evaluation):
It seems like the CR solution is important for learning a good dual solution. How does the learned dual solution compare with the CR dual solution in terms of GAP and GAP-CR?

Author response:
Yes the CR solution is important for learning a good dual solution.
As shown in ablation studies, a system without this information (-cr) cannot predict accurate LMs.

GAP-CR is exactly the metrics that  indicates how the learned dual solution compares with both the CR solution and the optimal solution.
- a negative score indicates the predicted solution is worse than the CR
- a zero score indicates that the solution is equivalent to CR
- a 100 score is reached when the solution is equivalent to the one returned by BM

Thank you for your interest. Clearly, we had not understood your question.
Using the dual variables $z^*$ associated with the dualized constraints as Lagrangian multipliers does not really improve the bound of the continuous relaxation CR. Here is a table indicating the CR gap closure given by $LR(z^*)$ on the different datasets (GAP-CR is equal to $(1 - \frac{LR(\pi^*) - LR(z^*)}{LR(\pi^*) - CR}) * 100$).

|    Dataset       |GAP-CR |
|:----------------:|------:|
| MCND-SMALL_COM40 | 0.16% |
| MCND-SMALL_COMVar| 0.06% |
| MCND-BIG-COM40   | 0.61% |
| MCND-BIG_COMVAR  | 0.28% |
| CFL              | 0.67% | 

We will add this value in the paper for comparison with our prediction.

Reviewer comment (Experiments):
It is unclear how the presented approach can really be beneficial. While experiments show that the network can predict good Lagrangian multipliers, enabling a subsequent bundle method to be warm started and its iteration count reduced by one third, it would have been essential to also compare running times to state-of-the-art IP solvers like Gurobi.

Author response:
It is not clear how to compare Lagrangian Relaxation with an IP solver (Gurobi or CPLEX) since they do not compute the same thing. Lagrangian Relaxation gives a dual bound. Comparing the time to obtain this bound with the time necessary to solve to optimality does not seem fair. Using cplex/gurobi to compute the linear relaxation is quite fast and this is why we use its solution as features to predict a better bound.

Reviewer comment (Reproducibility):
The paper does not provide the instances and the code as a supplement, which would allow reviewers to assess the work.

Author response:
Code and instances will be available upon the acceptation of the paper.

Reviewer comment (Experiments):
It is not clear how well the approach really learns to predict the multipliers. Given enough training samples, as in this work, it is important to consider how well a simple k-NN would perform.

Author response:
We ran experiments to compare our approach with supervised methods (small MCDN instances, here evaluated on the first fold only)  
Optimal lagrangians are given by SMS++:

| k   | GAP            | GAP-CR             |
| --- | -------------- | ------------------ |
| 3   | 58.12 (± 9.23) | -605.74 (± 527.38) |
| 5   | 54.08 (± 8.51) | -556.23 (± 489.36) |
| 10  | 52.32 (± 8.23) | -535.15 (± 473.33) | 

Clearly, simple k-NN is unable to retrieve interesting examples from the train data.

Moreover we compare our model with other learning objective.
We keep the encoder-decoder architecture of our model and change the objective for a supervised loss, either the Hinge Loss on the binary variable of each subproblem or more directly the MSE loss wrt to optimal LMs.
In both cases, the "optimal solutions" are given by SMS++ and the evaluation is performed over the small MCDN instances, here evaluated on the first fold only:

| Obj   | GAP  | GAP-CR |
| ----- | ---- | ------ |
| LR    | 1.90 | 85.12  |
| Hinge | 3.99 | 65.57  |
| MSE   | 9.10 | -3.16  |

Unsupervised learning (LR) is clearly better than supervised losses.
It should be noted that supervised learning is difficult, since there is an infinity of LM solutions that give the optimal bound.
Learning to imitate one solver, here SMS++, by predicting  the same LMs is not an efficient learning method. 

If we replace the GNN encoder/decoder with a MLP, as done in methods where all problems have the same size, and train with the same LR objective we obtain (again, we train/evaluate on small instances with 40 commodities), we get:

| Obj       | GAP  | GAP-CR |
| --------- | ---- | ------ |
| LR (full) | 1.90 | 85.12  |
| LR (MLP)  | 7.63 | 37.17  | 

GNNs are essential to force the representation of the dualized constraints to agree with each other.

Reviewer comment (Experiments):
Since MILPs are very important and the presented approach is very general, it would have been good to see it applied to more general and more common MILPs, as MCDN and CFL are somewhat special problems.

Author response:
We chose MCDN and CFL since they are well-known OR problems usually tackled by Lagrangian relaxation and satisfying the two mentionned requirements. On the contrary, the classic Lagrangian relaxation for TSP (one of the reference OR problems) based on one-tree decomposition does not satisfy any of our two requirements.

Reviewer comment (Evaluation):
How long does Gurobi need to solve the MILP instances?

Author response:
It is not clear how to compare Lagrangian Relaxation with an IP solver (Gurobi or CPLEX) since they do not compute the same thing. Lagrangian Relaxation gives a dual bound. Comparing the time to obtain this bound with the time necessary to solve to optimality does not seem fair. Using cplex/gurobi to compute the linear relaxation is quite fast and this is why we use its solution as features to predict a better bound.

Reviewer comment (Novelty):
How does the approach compare to other approaches that learn Lagrangian dual variables, as the paper states a number of such approaches for specific MILPs?

Author response:
Moreover we compare our model with other learning objective.
We keep the encoder-decoder architecture of our model and change the objective for a supervised loss, either the Hinge Loss on the binary variable of each subproblem or more directly the MSE loss wrt to optimal LMs.
In both cases, the "optimal solutions" are given by SMS++ and the evaluation is performed over the small MCDN instances, here evaluated on the first fold only:

| Obj   | GAP  | GAP-CR |
| ----- | ---- | ------ |
| LR    | 1.90 | 85.12  |
| Hinge | 3.99 | 65.57  |
| MSE   | 9.10 | -3.16  |

Unsupervised learning (LR) is clearly better than supervised losses.
It should be noted that supervised learning is difficult, since there is an infinity of LM solutions that give the optimal bound.
Learning to imitate one solver, here SMS++, by predicting  the same LMs is not an efficient learning method. 

If we replace the GNN encoder/decoder with a MLP, as done in methods where all problems have the same size, and train with the same LR objective we obtain (again, we train/evaluate on small instances with 40 commodities), we get:

| Obj       | GAP  | GAP-CR |
| --------- | ---- | ------ |
| LR (full) | 1.90 | 85.12  |
| LR (MLP)  | 7.63 | 37.17  | 

GNNs are essential to force the representation of the dualized constraints to agree with each other.

Reviewer comment (Novelty):
There is quite some literature on using DNN to improve combinatorial optimization, but the authors do not cite [2], which has a close connection with the work on the encoder-decoder refinement in training.

Author response:
Thank you for pointing out reference [2].  
We agree that it is a really great work that we could have cited in our paper. We did not mention it since our work is related to predicting dual bound, and not primal bound as in [2]. We cite instead (Nair et al.) when mentionning primal bound prediction as their model is on predicting primal bounds of any MILP, and not vehicle routing like in [2]. Moreover, we do not use Reinforcement learning as in [2].  
Still, we will take you feedback into account and cite it as a previous work on encoder-decoder architectures for optimization-based problems.

Reviewer comment (Experiments):
In the experimental section, the authors only conduct experiments on self-generated datasets, making it harder for outsiders to understand the results.

Author response:
We could not find datasets available for these problems, that is why we had to create new ones.  
Upon acceptance, we will release our datasets to encourage comparisons and further work.

Reviewer comment (Evaluation):
P8 on the bundle method warm start — does the time include the CR / DNN forward time?

Author response:
Q1/2: The results on page 8 indeed include the time for solving the continuous relaxation (CR) and making the prediction with our ML model. We will make it clearer in the revised version.