Handle shared constraints properly

[dfridovi]

We need to be careful about handling shared constraints. I am convinced that we need to enforce them at the lowest level of each player's nested optimization problem (if they are enforced, say, at only the top level, then they could easily be infeasible when considered alongside the nested optimality conditions that would be ignoring them). However, there is still a question of how to handle sharing multipliers for these constraints between players. There are two conflicting points of view:

1. Traditionally, multipliers are shared between players for shared constraints (apart from cases in which the players also share variables).
2. But, for a single player, we need to have different multipliers at each level that correspond to constraints enforced at the innermost level. This is critical for ensuring that we prioritize inner problem optimality.

So, the question is: at what level do the players share multipliers for shared constraints that must be enforced at each player's innermost level? Again, I see two naive possibilities:

1. Share the multipliers only at all players' innermost levels, but let successive outer levels have their own separate multipliers for all players.
2. Share the multipliers only at all players' outermost levels, and let successive inner levels have separate multipliers.

Thoughts, anyone?

[dfridovi]

I am leaning toward case (1) above, i.e. sharing multipliers only at the innermost levels which correspond to the highest priority preferences. The logic is that this will force players to pay the same "price" for enforcing the constraint where it matters most, and the separate multipliers at outer levels only serve to ensure we prioritize inner preferences anyway.

[lassepe]

I agree that it would be wrong to add the constraint only at the outer-most level but I would think it is sufficient to for them to be present at the inner-most level and then be carried up like all other constraints, no?

[dfridovi]

Right, but the subtlety is that to "carry them up" to higher priorities we need those higher levels to have their own multipliers (just like they need separate multipliers for inner private constraints). We did this in the original paper but the difference is we are now not treating inner duals as primals at outer levels (though inner lagrangian stationary conditions are given additional outer multipliers), and that really clamps down on the total number of variables (quadratic in depth vs exponential).