Enhancement

Implement a "Multiplier Model" in Cooper. This idea comes from the paper by Narasimhan et al.

Instead of tracking the Lagrange Multipliers of a constrained optimization problem explicitly, consider a model which takes as input features associated with each constraint and outputs their corresponding multiplier.

This requires a new kind of Multiplier in Cooper different from a DenseMultiplier. In turn, we might want to create and abstract class for multipliers outlining how a custom multiplier needs to be implemented.

This also requires a new formulation or the modification of an existing one. Note that the dual_parameters do not match the multipliers in this new functionality. Multipliers are the outputs of the Multiplier Model, whereas the dual_parameters would be the actual weights of said model.

Motivation

Consider problems with an extremely large (or even infinite) number of constraints. It becomes computationally intractable to evaluate and keep track of each constraint. Moreover, when solving the problem with an (explicit) Lagrangian formulation, there is the requirement to store and update an equally large number of multipliers.

With a Multiplier Model, Lagrange multipliers can be computed "on demand", given the constraints that were evaluated at each time step. As long as the Multiplier Model has less parameters than the total possible number of multipliers, there is a gain in terms of storage.

Also, note that the Multiplier Model's parameters are shared across constraints. Therefore, they can learn to have similar multipliers for constraints which tend to be similar to one another.

References

• H. Narasimhan, A. Cotter, Y. Zhou, S. Wang, and W. Guo. Approximate Heavily-Constrained Learning with Lagrange Multiplier Models. In NeurIPS, 2020b.