Implement bridge_cost for CachingOptimizer

[blegat]

Abstract description

Consider layers: LazyBridgeOptimizer(Dualization(LazyBridgeOptimizer)) (with possible caching layers in between).
If you have a set S that does not implement dual_set but that has bridges, then a constraint F-in-S with this set are not supported by the dualization layer so it will be bridged by the outer layer. Now, all the other constraints that ##are supported by dualization will just go through the dualization layer will bridging cost of 0, even though they may be bridged by the inner bridges. Because of this, the bridge choice for F-in-S will be completely blind and might be sub-optimal.

Concrete example

The whole idea detailed in jump-dev/SumOfSquares.jl#205 explains that we have a KernelBridge and an ImageBridge and the right one will be picked thanks to the bridge cost

Written by Claude, I haven't checked yet. I have now iterated, looks good.

[odow]

Missing a test when the CachingOptimizer has NO_OPTIMIZER.

I'll run solver-tests:

https://github.com/jump-dev/MathOptInterface.jl/actions/runs/25891420946

[odow]

This breaks a lot of packages

[blegat]

Looking at the breaking, some are already failing: https://github.com/jump-dev/MathOptInterface.jl/actions/runs/25967424335
For the other ones, it seems that layers were incorrectly implementing the bridging cost attributes.

• Implement bridge_cost Dualization.jl#227
• Fix bridging cost ParametricOptInterface.jl#240
• Fix bridging cost for QCQP PolyJuMP.jl#146
• Fix ConstraintBridgeCost Convex.jl#729

We should release patch version of these packages before releasing this but then it should be fine. It might break MOI layers we are not testing with solver-tests though :/
The thing is that they are not implementing the bridging cost correctly so in a sense we're catching an existing bug.
What we could do is first to a release that adds tests to catch such bugs.