Dual Sign Changes Explanations

[andrewrosemberg]

We are trying to understand the dual sign changes in the Ipopt wrapper.

I.e., How the dual multiplier

Ipopt.jl/src/MOI_wrapper.jl

Line 1325 in 4c15646

_dual_multiplier(model::Optimizer) = model.sense == MOI.MIN_SENSE ? 1.0 : -1.0

is used depending on each constraint / bound type: e.g.,

Ipopt.jl/src/MOI_wrapper.jl

Line 1340 in 4c15646

s = -_dual_multiplier(model)

Can we have some line comments for each case explaining the reasoning?

[odow]

All MOI.AbstractOptimizers implement the same dual convention: https://jump.dev/MathOptInterface.jl/stable/background/duality/

• The dual of a >= (MOI.GreaterThan / MOI.Nonnegatives) constraint is non-negative, because the dual cone of MOI.Nonnegatives is MOI.Nonnegatives.
• The dual of a <= (MOI.LessThan / MOI.Nonpositives) constraint is non-positive, because the dual cone of MOI.Nonpositives is MOI.Nonpositives
• The dual of a == constraint is free, and the sign depends on which "side" of the equality is binding (see two points above)

It has been a while since I needed to remember the details of Ipopt, but from memory, the dual of the constraint rows (.mult_g) had the opposite convention to the variable bounds (.mult_x_L and .mult_x_U)?

I remember it being confusing.

[odow]

Closing because this is not a bug in Ipopt. (But please comment if you have an example of a bug where the above doesn't hold.)