[Bridges] fix adjoint functions in SetDotInverseScalingBridge

[odow]

Potential fix for #2262 (comment).

I don't really understand what's going here, but it makes sense that the adjoint should be the opposite of the forward map? That's what it is for SetDotScalingBridge, so I think this is just a typo.

Seemed to fix SDPA, so I'll re-run solver-tests on this branch: https://github.com/jump-dev/MathOptInterface.jl/actions/runs/6103555292

[odow]

Mosek still broken, but because you explicitly coded the bridge type into a test:
https://github.com/jump-dev/MosekTools.jl/blob/6624a98dcef6b4d6619111e9b40b8afb05b40bae/test/runtests.jl#L264

[blegat]

This is weird. The adjoint is the transpose and the transpose if a Diagonal should be the matrix itself

[blegat]

We should add a test that sets and gets ConstraintDualStart in the bridge test, it should clarify

[odow]

We should add a test that sets and gets ConstraintDualStart in the bridge test, it should clarify

There is a test. But it just tests that the round trip preserves. So it doesn't help if it gets the wrong value... 😆

[odow]

This is weird. The adjoint is the transpose and the transpose if a Diagonal should be the matrix itself

Isn't the SetDotInverseScalingBridge just the inverse of this? So it makes sense they'd be swapped.

MathOptInterface.jl/src/Bridges/Constraint/bridges/set_dot_scaling.jl

Lines 90 to 108 in fdf2559

	# Since the map is a diagonal matrix `D`, it is symmetric so one would initially
	# expect `adjoint_map_function` to be the same as `map_function`. However, the
	# scalar product for the scaled PSD cone is `<x, y>_2 = x'y` but the scalar
	# product for the PSD cone additionally scales the offdiagonal entries by `2`
	# hence by `D^2` so `<x, y>_1 = x'D^2y`.
	# So `<Dx, y>_2 = <x, D^(-1)y>_1` hence the adjoint of `D` is its inverse!
	function MOI.Bridges.adjoint_map_function(
	::Type{<:SetDotScalingBridge{T,S}},
	func,
	) where {T,S}
	return _inverse_scale(T, S, func)
	end
	function MOI.Bridges.inverse_adjoint_map_function(
	::Type{<:SetDotScalingBridge{T,S}},
	func,
	) where {T,S}
	return _scale(T, S, func)
	end

[blegat]

It's good that I left a comment to explain this to my future self 😆 . So this fix is indeed correct because the adjoint is not the transpose. Can you add a test that don't just check the round trip ?