# How can you get the Lagrange/KKT multipliers out of JuMP when using Ipopt?

**URL:** <https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432>\
**Category:** Optimization (Mathematical)\
**Tags:** jump\
**Created:** [December 4, 2024, 1:09am UTC](https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432 "2024-12-04T01:09:50Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![lamont](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lamont/32/27486_2.png) [@lamont](https://discourse.julialang.org/u/lamont)\
**Post date:** [December 4, 2024, 1:09am UTC](https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432/1 "2024-12-04T01:09:50Z")

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CasADI has lam\_x and lam\_g outputs. I can’t find in what they’re calling in Ipopt to get these, or I’d try digging into the Ipopt.jl code to see where it was. I haven’t had any luck with the jump-dev or ipopt documentation or web searches…

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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:** [December 4, 2024, 1:32am UTC](https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432/2 "2024-12-04T01:32:49Z")

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JuMP calls these “dual” solutions: [Solutions · JuMP](https://jump.dev/JuMP.jl/stable/manual/solutions/#Dual-solutions)

```Julia
julia> using JuMP

julia> using Ipopt

julia> model = Model(Ipopt.Optimizer)
A JuMP Model
├ solver: Ipopt
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, x >= 0)
x

julia> @variable(model, 0 <= y <= 3)
y

julia> @objective(model, Min, 12x + 20y)
12 x + 20 y

julia> @constraint(model, c1, 6x + 8y >= 100)
c1 : 6 x + 8 y ≥ 100

julia> @constraint(model, c2, 7x + 12y >= 120)
c2 : 7 x + 12 y ≥ 120

julia> optimize!(model)
This is Ipopt version 3.14.16, running with linear solver MUMPS 5.7.3.

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 4
Number of nonzeros in Lagrangian Hessian.............: 0

Total number of variables............................: 2
                     variables with only lower bounds: 1
                variables with lower and upper bounds: 1
                     variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 2
        inequality constraints with only lower bounds: 2
   inequality constraints with lower and upper bounds: 0
        inequality constraints with only upper bounds: 0

iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
   0 3.1999968e-01 1.20e+02 7.60e-01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 4.7893512e+00 1.17e+02 1.42e+01 -1.0 3.44e+01 - 1.82e-03 2.88e-02h 1
   2 1.9573531e+01 1.08e+02 1.32e+01 -1.0 1.19e+01 - 9.92e-03 7.08e-02h 1
   3 1.3373464e+02 4.17e+01 5.40e+00 -1.0 1.43e+01 - 1.00e+00 6.18e-01h 1
   4 2.0522904e+02 0.00e+00 2.09e-02 -1.0 5.55e+00 - 7.19e-01 1.00e+00h 1
   5 2.0503895e+02 0.00e+00 2.00e-07 -1.7 4.52e-01 - 1.00e+00 1.00e+00f 1
   6 2.0500506e+02 0.00e+00 2.83e-08 -2.5 6.69e-02 - 1.00e+00 1.00e+00f 1
   7 2.0500030e+02 0.00e+00 1.50e-09 -3.8 7.95e-03 - 1.00e+00 1.00e+00f 1
   8 2.0500000e+02 0.00e+00 1.84e-11 -5.7 5.95e-04 - 1.00e+00 1.00e+00f 1
   9 2.0500000e+02 0.00e+00 2.71e-14 -8.6 7.38e-06 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 9

                                   (scaled) (unscaled)
Objective...............: 2.0499999795501228e+02 2.0499999795501228e+02
Dual infeasibility......: 2.7149984354261548e-14 2.7149984354261548e-14
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5088486144185382e-09 2.5088486144185382e-09
Overall NLP error.......: 2.5088486144185382e-09 2.5088486144185382e-09

Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 1
Number of Lagrangian Hessian evaluations = 1
Total seconds in IPOPT = 0.001

EXIT: Optimal Solution Found.

julia> dual(c1)
0.25000000012313955

julia> dual(c2)
1.499999999870584

julia> dual(LowerBoundRef(x))
1.6710061615730542e-10

julia> dual(LowerBoundRef(y))
5.678764080540814e-10

julia> dual(UpperBoundRef(y))
0.0

```

Note that there may be a factor of `-1` in there, depending on the convention you expect. JuMP’s convention is: the dual of `>=` is non-negative, and the dual of `<=` is non-positive.

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [December 4, 2024, 1:55am UTC](https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432/3 "2024-12-04T01:55:31Z")

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With any optimization software, you can always get the Lagrange multipliers / dual variables from the objective/constraint gradients at the optimum just by solving a linear equation (from KKT).

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**Author:** ![lamont](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lamont/32/27486_2.png) [@lamont](https://discourse.julialang.org/u/lamont)\
**Post date:** [December 4, 2024, 3:46am UTC](https://discourse.julialang.org/t/how-can-you-get-the-lagrange-kkt-multipliers-out-of-jump-when-using-ipopt/123432/4 "2024-12-04T03:46:46Z")

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Thanks, that makes perfect sense in retrospect, I just hadn’t hit on using that search term.