# To confirm: The sign of the dual variable on a constraint of \`JuMP\` model

**URL:** <https://discourse.julialang.org/t/to-confirm-the-sign-of-the-dual-variable-on-a-constraint-of-jump-model/114292>\
**Category:** Optimization (Mathematical)\
**Tags:** question, jump, optimization\
**Created:** [May 15, 2024, 9:33am UTC](https://discourse.julialang.org/t/to-confirm-the-sign-of-the-dual-variable-on-a-constraint-of-jump-model/114292 "2024-05-15T09:33:42Z")\
**Posts on this page:** 1\
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**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [May 15, 2024, 10:12am UTC](https://discourse.julialang.org/t/to-confirm-the-sign-of-the-dual-variable-on-a-constraint-of-jump-model/114292/2 "2024-05-15T10:12:41Z")

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You’re right that that can be a confusing topic. We should probably add more explanation to the JuMP docs.

A key realization is that:

- The sign convention is arbitrary. I could assume that \lambda was non-positive if I put a - sign in front of the summation
- JuMP uses the duality conventions from conic duality.

These part of the documentation may be helpful:

- [Constraints · JuMP](https://jump.dev/JuMP.jl/stable/manual/constraints/#constraint_duality)
- [Duality · MathOptInterface](https://jump.dev/MathOptInterface.jl/stable/background/duality/)

The way I remember `JuMP.dual` is:

- the sign of `JuMP.dual(constraint)` does not depend on whether we are minimizing or maximizing
- the sign of `JuMP.dual(constraint)` is non-negative for \ge and non-positive for \le

In more detail:

- the dual of a \le constraint is non-positive, because f(x) \le y is rewritten to f(x) - y \in \mathbb{R}\_- (the `MOI.Nonpositives` cone), and the dual cone of `MOI.Nonpositives` is `MOI.Nonpositives`
- the dual of a \ge constraint is non-negative, because f(x) \ge y is rewritten to f(x) - y \in \mathbb{R}\_+ (the `MOI.Nonnegatives` cone), and the dual cone of `MOI.Nonnegatives` is `MOI.Nonnegatives`.
- the dual of a = constraint is free, because f(x) = y is rewritten to f(x) - y \in {0} (the `MOI.Zeros` cone), and the dual of the `MOI.Zeros` cone is `MOI.Reals`

Notably, this is the opposite to your statement that `λ is required to be nonnegative`, so to convert to your assumed Lagragian, you need to add `-` sign.

`shadow_price` and `reduced_cost` are mostly useful for people coming from a background in linear programming. There, `shadow_price` is the change in the objective value as the constraint is relaxed, and the sign depends on whether we are maximizing or minimizing.

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