Implement Feasibility Cuts for PlasmoBenders

[dlcole3]

Currently, PlasmoBenders does not handle infeasible solutions very well. There is an option to add slacks to the linking constraints, but in practice, they do not have the greatest performance. It would be helpful to try implementing feasibility cuts as an alternative

[odow]

Linking to this section of the JuMP docs: https://jump.dev/JuMP.jl/stable/tutorials/algorithms/benders_decomposition/#Feasibility-cuts

The JuMP trick is to recognize that you can use dual_objective_value(model) when there is an infeasibility certificate for the intercept.

[dlcole3]

Feasibility cuts have been implemented for two-stage problems in #20. For problems with 3+ stages, I need to update the algorithm so that it terminates after solving for the given stage where the infeasibility occurred and then start a new pass of the algorithm. The challenge here is that:

• the upper and lower bound computations will need to be updated for 3+ stage problems
• if the forward pass has an infeasibility, the backwards pass needs to only continue up to whatever stage had the infeasibility
• the other subproblems in a given stage should also be solved and cuts passed backwards
  I think one way of addressing this is to make the algorithm "step through" the graph. Right now, I just loop over all subproblems (which are ordered by stage). Instead, I should use the next_objects dictionary I created to know which problems to solve next and probably have a running queue.

[odow]

I think one way of addressing this is to make the algorithm "step through" the graph

This is the way. See https://sddp.dev/stable/explanation/theory_intro/ for an example (optimality only, no feasibility)