MultiJuMP enables the user to easily run multiobjective optimisation problems and generate Pareto fronts.
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Updated Last
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Started In
September 2015

JuMP v1.8 update

JuMP v1.8.0 added native support for multiobjective problems, so there is no longer a need to use the MultiJuMP.jl extension. See the JuMP documentation for more details:

A collection of solution algorithms is available in the MultiObjectiveAlgorithms.jl package: Please open an issue there if you have feature requests or bug reports.


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MultiJuMP enables the user to easily run multiobjective optimisation problems and generate Pareto fronts. The code is built as an extension of JuMP. We have implemented three ways to trace out the Pareto front:

  • Normal Boundary Intersection (optimize!(m, method = NBI()))
    • This method is applicable only for nonlinear optimisation
  • Weighted sums (optimize!(m, method = WeightedSum()))
  • Constraint methods (optimize!(m, method = EpsilonCons()))
    • This method only works for biobjective optimisation as of now

Disclaimer: MultiJuMP is not developed or maintained by the JuMP developers.


In Julia, call Pkg.add("MultiJuMP") to install MultiJuMP.


Have a look at the tests and examples/ directory for different use cases, including tri-objective Pareto fronts.

MultiJuMP supports linear problems using the linear=true keyword when calling multi_model(linear=true). Currently, only the EpsilonCons() and WeightedSum() methods are supported for linear problems.

using MultiJuMP, JuMP
using Clp

const mmodel = multi_model(Clp.Optimizer, linear = true)
const y = @variable(mmodel, 0 <= y <= 10.0)
const z = @variable(mmodel, 0 <= z <= 10.0)
@constraint(mmodel, y + z <= 15.0)

# objectives
const exp_obj1 = @expression(mmodel, -y +0.05 * z)
const exp_obj2 = @expression(mmodel, 0.05 * y - z)
const obj1 = SingleObjective(exp_obj1)
const obj2 = SingleObjective(exp_obj2)

# setting objectives in the data
const multim = get_multidata(mmodel)
multim.objectives = [obj1, obj2]

optimize!(mmodel, method = WeightedSum())

# Get the Utopia and Nadir points
utopiapoint = getutopia(multim)
nadirpoint = getnadir(multim)

Plotting with Plots.jl is supported via recipes:

using Plots: plot, title!, scatter!
pltlin = plot(multim)
title!(pltlin, "Extrema of the Pareto front")

# Show Utopia and Nadir points
# (This is probably a hacky way to do this)
    [utopiapoint[1], nadirpoint[1]], [utopiapoint[2], nadirpoint[2]],

Linear pareto front

As a non-linear usage example, we implement the test from Das and Dennis, 1998: Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems:

using MultiJuMP, JuMP
using Ipopt

m = multi_model(Ipopt.Optimizer)
@variable(m, x[i=1:5])
@NLexpression(m, f1, sum(x[i]^2 for i=1:5))
@NLexpression(m, f2, 3x[1]+2x[2]-x[3]/3+0.01*(x[4]-x[5])^3)
@NLconstraint(m, x[1]+2x[2]-x[3]-0.5x[4]+x[5]==2)
@NLconstraint(m, 4x[1]-2x[2]+0.8x[3]+0.6x[4]+0.5x[5]^2 == 0)
@NLconstraint(m, sum(x[i]^2 for i=1:5) <= 10)

iv1 = [0.3, 0.5, -0.26, -0.13, 0.28] # Initial guess
obj1 = SingleObjective(f1, sense = MOI.MIN_SENSE,
                       iv = Dict{String,Any}("x[$i]" => iv1[i] for i in 1:length(iv1)))
obj2 = SingleObjective(f2, sense = MOI.MIN_SENSE)

md = get_multidata(m)
md.objectives = [obj1, obj2]
md.pointsperdim = 20
optimize!(m, method = NBI(false)) # or method = WeightedSum() or method = EpsilonCons()

# Get the Utopia and Nadir points
utopiapoint = getutopia(md)
nadirpoint = getnadir(md)

using Plots
pltnbi = plot(md)

NBI Pareto front example

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