Summary
This paper addresses the broad category of machine learning (ML) for optimization, where ML-based approaches are used to obtain solutions or bounds on the solutions of optimization problems. In particular, the paper addresses a particular subclass of problem - conic problems, and even more specifically, the dual solution of these problems.
Most "ML-for-optimization" frameworks have thus far focused on finding the solution to primal problems, which is intuitive because the variables in primal problems are typically the variables of interest for most real-world problems. However, learning dual solutions can help certify or bound the primal solution, and in some cases, one may be interested in the dual solution directly (for example, when learning shadow prices in economic optimization problems). This paper uses a conic projection layer in addition to the dual completion step to ensure dual feasibility.
The paper compares the method (DLL) to other existing methods in terms of their ability to learn solutions to dual problems and shows superiority, mostly due to the fact that DLL includes dual optimality conditions in addition to dual feasibility constraints. An alternative to Euclidean projection is also provided (Radial projection), for the projection layer, which allows for computation once per cone and attempts to avoid gradient vanishing issues during training.
Strengths
The paper claims that the model can guarantee dual feasibility, which is significant for yielding bounds on the primal solution. In addition, the authors show a closed-form, analytical solution for conic projections and for dual conic completion, outside of the ML model development.
The proposed DLL framework doesn't require an explicit formulation of the dual problem; instead, the user must only have to provide a set of primal constraints that satisfy a certain set of conditions. This is convenient but not a main contribution (if the primal conic problem is known, the dual should be relatively easy to find).
Even though the most challenging and time consuming optimization problems in practice are nonconvex optimization problems, many of these problems can be cast as conic problems, and thus the method can be applied to solve quite a large array of problems that may occur in practice.
Versus other methods, the DLL framework requires far less hyperparameter tuning and iterative procedures, which contributes to the ease of training and increased computational speed.
Weaknesses
One limitation of the method, if a known closed-form optimal dual completion cannot be achieved, is its use of implicit layers. It would be interesting to see how the speedups diminish with very large problems where the implicit layer becomes a bottleneck.
The comparison with existing approaches is fair since those are the state of the art for learning primal solutions, but it's slightly unfair because those methods were not specifically designed to learn dual solutions. I think this is fine since there are not existing ML-for-optimization methods tailored towards dual solutions in particular, but just to note.
The work in this paper does seem like a more generalized and extended version of the work in reference [19], where reference [19] was the first to provide a dual proxy for a second-order cone relaxation of the AC optimal power flow (OPF) problem. However, in reading [19], it does not seem to be necessarily limited to the AC OPF problem, although those authors only apply it to such. The approach in the present paper is generalized to a broader class of problems, but is not the first to provide dual feasibility guarantees as determined by a learning model.
Questions
It would be interesting to also see indications of whether or not strong duality holds in the test cases, if this is not too challenging to check. If so, the optimality gaps reported also should hold in the primal. The authors may also want to come up with some examples of why learning dual solutions directly may be a contribution on its own, outside of just providing a bound on a primal solution.
I would also want to see more of a discussion about the comparison with reference [19], specifically, is [19] already generalizable but the authors only applied it to the AC OPF problem in the paper?
Regarding the use of implicit layers, a discussion (or even better, experiments if time allows) on their computational limitations and how this slows down inference as problem size grows
Limitations
The authors have sufficiently addressed the limitations of the method. I don't see any potential negative societal impacts of the work.