# JuMP.shadow\_price conveys a misconception?

**URL:** <https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, dual\
**Created:** [March 30, 2025, 4:05am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503 "2025-03-30T04:05:21Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 30, 2025, 4:05am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/1 "2025-03-30T04:05:21Z")

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> In constraint optimization, the **shadow price** is the change, **per infinitesimal unit** of the constraint, in the optimal value of the objective function, obtained by **relaxing** the constraint.

This is the wikipedia definition of the **shadow price** , which is also adopted by JuMP.  
We can infer from it that shadow price is a primal-side quantity which is **unrelated to the dual side**.

Let’s see this LP example, in which `3 * y >= x`, namely `μ` is the constraint that we aim to study

```julia
primal = JuMP.Model(Gurobi.Optimizer)
JuMP.@variable(primal, 0 <= y); JuMP.@variable(primal, 0 <= x)
JuMP.@objective(primal, Min, 2 * y + 0.7 * x)
JuMP.@constraint(primal, μ, 3 * y >= x)
# For ease of reference
# The dual program of `primal` is the following 4 lines
# Max 0
# -3 μ >= -2 [⇐ y ≥ 0]
# μ >= -0.7 [⇐ x ≥ 0]
# μ >= 0

```

If we want to derive the shadow price **by definition** , we should add a parameter `p` and update the constraint to `3 * y + p >= x`. Here `p` should be a small positive number, where “small” is due to the defining word “ **infinitesimal** ”, and “positive” is due to the defining word “ **relaxing** ”.  
Considering the updated LP, we conclude that

1. The optimal objective value can never be negative.
2. when `x = y = 0`, the primal problem is feasible with objective attaining `0`

Therefore, the optimal objective value (after or before perturbed by `p`) is always `0`.  
Hence, the difference is `0`. Therefore, after get divided by `p`, it is still `0`.

We conclude that the **shadow price** of the `μ` constraint is `0`.

Nevertheless, the `shadow_price` function in JuMP is associated to the `dual`—a totally distinct concept. Secondarily, since the `JuMP.dual` function depends on the solver, it is subject to numeric precision, thereby cannot return a precise “change rate”.

The following runnable code shows that JuMP may provide a false result (`🍅`).  
(false means being inconsistent with the aforementioned **definition** )

```julia
import JuMP, Gurobi
model = JuMP.Model(Gurobi.Optimizer)
JuMP.@variable(model, 0 <= y); JuMP.@variable(model, 0 <= x)
JuMP.@objective(model, Min, 2 * y + 0.7 * x)
JuMP.@constraint(model, μ, 3 * y >= x)
JuMP.optimize!(model); JuMP.assert_is_solved_and_feasible(model; allow_local = false, dual = true)
@assert JuMP.shadow_price(μ) == -1 * JuMP.dual(μ)
JuMP.shadow_price(μ) # -0.6666666666666666 🍅

```

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<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 30, 2025, 4:17am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/2 "2025-03-30T04:17:08Z")

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Put simply, the following 2 statements (`🔵 and 🔴`) are inconsistent

```julia
help?> JuMP.shadow_price
  shadow_price(con_ref::ConstraintRef)

  🔵Return the change in the objective from an infinitesimal relaxation of the constraint.

  🔴The shadow price is computed from dual and can be queried only when the objective sense is     
  MIN_SENSE or MAX_SENSE (not FEASIBILITY_SENSE).

```

`🔵` adopts the wikipedia definition, which is purely primal-side.  
`🔴` depends wholly on the dual-side, and is thereby another concept.

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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:** [March 30, 2025, 9:05am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/3 "2025-03-30T09:05:13Z")

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The shadow price is defined exactly in terms of the dual variables (really just flipping the sign in some cases). They don’t represent different concepts.

Your confusion stems from the fact that your example has a redundant constraint/the primal is degenerate/there are multiple optimal dual solutions.

To see this, try writing the primal as:

```julia
primal = Model()
@variable(primal, x)
@variable(primal, y)
@objective(primal, Min, 2 * y + 0.7 * x)
@constraint(primal, c1, 3 * y >= x)
@constraint(primal, c2, x >= 0)
@constraint(primal, c3, y >= 0)

```

The single sentence definition of the shadow price is the commonly used definition, and sure, it doesn’t account for some of the edge-cases that can happen. This gets back at something we discussed a few days ago: to what extent should the documentation of JuMP also be a textbook on general optimization concepts?

I don’t know that the docstring of `shadow_price` needs a lot of detail. Shadow price is a commonly used term of art in the field, and people can refer to other sources for the various technical aspects.

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<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 30, 2025, 10:40am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/4 "2025-03-30T10:40:46Z")

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> [@odow](#):
>
> your example has a redundant constraint

I wrote a [note](https://discourse.julialang.org/t/a-note-on-how-to-identify-redundant-constraints-using-jump/127504) on this to facilitate future reference

> [@odow](#):
>
> I don’t know that the docstring of `shadow_price` needs a lot of detail.

About that docstring, I think at least we can swap its 2nd and 1st statement.

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<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [June 11, 2025, 5:33am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/5 "2025-06-11T05:33:18Z")

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> [@odow](#):
>
> /the primal is degenerate/

Could you be a bit more specific about this concept? in aspects including

1. is the LP problem degenerate, or is a solution to that problem degenerate?
2. is the concept of degenerate related to the objective function of an LP?
3. are you aware of another concept named “dual degeneracy”?

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<div class="post-metadata">

**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:** [June 11, 2025, 7:44am UTC](https://discourse.julialang.org/t/jump-shadow-price-conveys-a-misconception/127503/6 "2025-06-11T07:44:19Z")

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I suggest that you take a look at any standard textbook. The keywords to search for are “linear program” and “degenerate basic solution”.