# JuMP: the year in review (2023)

milestones ·Author: Oscar Dowson

At the end of last year, I gave a talk at the 55th Operations Research Society of New Zealand annual conference (slides). It proved to be a good excuse to summarize the progress we made in JuMP over the past year, and, since we didn’t record the talks, the President of ORSNZ, Andrea Raith, suggested a blog post might be in order.

Each of the following sections is a brief summary of a part of JuMP that we improved over the past year.

- About JuMP
- Nonlinear
- Complementarity
- Multi-objective
- Complex numbers
- Generic numbers
- Constraint programming
- Time-to-first-solve
- Future plans

## About JuMP

JuMP is a modeling language and collection of supporting packages for mathematical optimization in Julia.

JuMP makes it easy to formulate and solve a range of problem classes, including linear programs, integer programs, conic programs, semidefinite programs, and constrained nonlinear programs.

People use JuMP to model and solve real-world problems, including large-scale inventory routing problems at Renault, train scheduling at Thales Inc., and school bus routing for Boston Public Schools.

A major use-case of JuMP is building continental-scale energy system models, including MIT’s GenX, the National Renewable Energy Laboratory’s Sienna, Spine’s SpineOpt.jl, and TNO’s TulipaEnergyModel.jl

In academia, people use JuMP to teach optimization at universities around the world, including DTU, École des Ponts, KAIST, MIT, PUC-Rio, Université de Nantes, and U. Wisconsin-Madison.

## Nonlinear

I spent the past two years working with Carleton Coffrin and colleagues at Los Alamos National Laboratory on a project to rewrite JuMP’s nonlinear syntax.

Despite the amount of work behind the scenes over two years, the changes to
user-facing code may seem relatively trivial: instead of using the various `@NL`

macros, you can now create and use nonlinear expressions in the regular JuMP
macros, for example:

```
julia> using JuMP
julia> model = Model();
julia> @variable(model, x[1:2]);
julia> @objective(model, Min, exp(x[1]) - sqrt(x[2]))
exp(x[1]) - sqrt(x[2])
```

Previously unsupported features, such as array operations and broadcasting, now work just like they do for linear and quadratic expressions:

```
julia> @expression(model, log(sum(exp.(x))))
log(0.0 + exp(x[2]) + exp(x[1]))
```

The JuMP documentation has more examples of the new syntax.

Behind the scenes, a lot of work went in to identify a suitable data structure for representing the nonlinear expressions, to refactor the various parts of JuMP and MathOptInterface, and to update solvers like Ipopt to the new syntax. You can find out more of the details by watching talks I gave at JuMP-dev in 2022 and 2023, both helpfully called the same thing:

- Improving nonlinear programming support in JuMP (2022)
- Improving nonlinear programming support in JuMP (2023)

## Complementarity

As part of the nonlinear work, we also extended JuMP to support nonlinear complementarity problems. This addressed a long-standing feature request for JuMP, which previously supported only linear and quadratic complementarity constraints.

As an example, you can now build and solve complementarity problems that look like this:

```
julia> using JuMP, PATHSolver
julia> function solve_insurance_problem(; pi = 0.01, L = 0.5, γ = 0.02, ρ = -1)
U(C) = -1 / C
MU(C) = 1 / C^2
model = Model(PATHSolver.Optimizer)
set_silent(model)
@variable(model, EU, start = 1) # Expected utility
@variable(model, EV, start = 1) # Equivalent variation in income
@variable(model, C_G, start = 1) # Consumption on a good day
@variable(model, C_B, start = 1) # Consumption on a bad day
@variable(model, K, start = 1) # Coverage
@constraints(model, begin
(1 - pi) * U(C_G) + pi * U(C_B) - EU ⟂ EU
100 * (((1 - pi) * C_G^ρ + pi * C_B^ρ)^(1 / ρ) - 1) - EV ⟂ EV
1 - γ * K - C_G ⟂ C_G
1 - L + (1 - γ) * K - C_B ⟂ C_B
γ * ((1 - pi) * MU(C_G) + pi * MU(C_B)) - pi * MU(C_B) ⟂ K
end)
optimize!(model)
return value(K)
end
solve_insurance_problem (generic function with 1 method)
julia> solve_insurance_problem()
0.20474003534537757
```

The JuMP documentation has a number of other examples that demonstrate the new syntax.

The new complementarity support is helping others in the community. David Anthoff and colleagues at UC Berkeley are working on MPSGE.jl, a reimplementation of Tom Rutherford’s MPSGE from GAMS into JuMP. Tom’s group is also looking at adopting JuMP for their official WiNDC model.

## Multi-objective

This year saw the introduction of support for multi-objective programs. There have been a few previous attempts at this, most notably Xavier Gandibleux’s vOptSolver project, and the MultiJuMP.jl extension.

You can now define a multi-objective program by passing a vector of scalar objective functions:

```
julia> model = Model();
julia> @variable(model, x[1:2]);
julia> @objective(model, Min, [1 + x[1], 2 * x[2]])
2-element Vector{AffExpr}:
x[1] + 1
2 x[2]
```

We also developed the MultiObjectiveAlgorithms.jl (MOA) package, which implements a number of solution algorithms for multi-objective programs.

The optimizer returns a set of points from the efficient frontier (exactly what
and how many depend on the solution algorithm), which you can access with the
`result`

keyword argument to functions like `JuMP.value`

and `JuMP.objective_value`

.

An example is:

```
julia> using JuMP, HiGHS
julia> import MultiObjectiveAlgorithms as MOA
julia> function solve_biobjective_assignment()
C1 = [5 1 4 7; 6 2 2 6; 2 8 4 4; 3 5 7 1];
C2 = [3 6 4 2; 1 3 8 3; 5 2 2 3; 4 2 3 5];
n = size(C2, 1)
model = Model(() -> MOA.Optimizer(HiGHS.Optimizer));
set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint());
set_silent(model)
@variable(model, x[1:n, 1:n], Bin);
@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
@constraint(model, [i = 1:n], sum(x[i, :]) == 1);
@constraint(model, [j = 1:n], sum(x[:, j]) == 1);
optimize!(model)
pair(a) = a.I[1] => a.I[2]
for i in 1:result_count(model)
sol = pair.(findall(>(0.5), value.(x; result = i)))
println(i, ": ", sol, " | ", objective_value(model; result = i))
end
end
solve_biobjective_assignment (generic function with 1 method)
julia> solve_biobjective_assignment()
1: [3 => 1, 1 => 2, 2 => 3, 4 => 4] | [6.0, 24.0]
2: [3 => 1, 2 => 2, 1 => 3, 4 => 4] | [9.0, 17.0]
3: [1 => 1, 2 => 2, 3 => 3, 4 => 4] | [12.0, 13.0]
4: [4 => 1, 2 => 2, 3 => 3, 1 => 4] | [16.0, 11.0]
5: [2 => 1, 4 => 2, 1 => 3, 3 => 4] | [19.0, 10.0]
6: [2 => 1, 4 => 2, 3 => 3, 1 => 4] | [22.0, 7.0]
```

Credit goes to Gökhan Kof for contributing many of the more advanced algorithms in MOA.jl.

## Complex numbers

Another big change over the last year or so was Benoît Legat’s work on adding complex number support to JuMP. This is a commonly requested feature that often arises in models such as power systems.

You can now use the `ComplexPlane`

syntax to add a complex-valued decision
variable to a JuMP model and use it in complex-valued equality constraints:

```
julia> using JuMP
julia> model = Model();
julia> @variable(model, x in ComplexPlane())
real(x) + imag(x) im
julia> @constraint(model, (1 + 2im) * x == 4 + 5im)
(1 + 2im) real(x) + (-2 + im) imag(x) = (4 + 5im)
```

At present, JuMP implements the rectangular formulation, where two real-valued decision variables are added to the model, one each for the real and imaginary components.

You can also create Hermitian matrices via:

```
julia> @variable(model, H[1:3, 1:3] in HermitianPSDCone());
```

See the documentation for more details.

## Generic numbers

In addition to complex numbers, Benoît also added a new `GenericModel{T}`

constructor, which creates a JuMP model in which the number type is `T`

instead
of the default `T = Float64`

.

As one example, you can now build and solve linear programs in exact rational arithmetic:

```
julia> using JuMP, CDDLib
julia> model = GenericModel{Rational{BigInt}}(CDDLib.Optimizer{Rational{BigInt}});
julia> @variable(model, 1 // 7 <= x[1:2] <= 2 // 3)
2-element Vector{GenericVariableRef{Rational{BigInt}}}:
x[1]
x[2]
julia> @constraint(model, c1, (2 // 1) * x[1] + x[2] <= 1);
julia> @constraint(model, c2, x[1] + 3x[2] <= 9 // 4);
julia> @objective(model, Max, sum(x));
julia> optimize!(model)
julia> value.(x)
2-element Vector{Rational{BigInt}}:
1//6
2//3
julia> objective_value(model)
5//6
```

See the documentation for more details.

## Constraint programming

The nonlinear improvements and generic number support combine to enable some nice improvements to constraint programming. Although JuMP isn’t a fully fledged constraint programming language like MiniZinc, with help from Chris Coey, it is now possible to build and solve models like this:

```
julia> using JuMP, MiniZinc
julia> model = GenericModel{Int}(() -> MiniZinc.Optimizer{Int}("chuffed"));
julia> @variable(model, 1 <= x[1:3] <= 3);
julia> @constraint(model, x in MOI.AllDifferent(3));
julia> @constraint(model, (x[1] == 2 || x[3] == 2) == true)
((x[1] == 2) || (x[3] == 2)) - 1 = 0
julia> @objective(model, Max, sum(i * x[i] for i in 1:3));
julia> optimize!(model)
julia> value.(x)
3-element Vector{Int64}:
2
1
3
```

## Time-to-first-solve

Another thing that improved dramatically over the past year was the infamous “time-to-first-solve” problem. This is the latency that users experience the first time they solve a JuMP model in a new Julia session.

In Julia 1.6 and earlier, it was common for this latency to be greater than 15 seconds. On Julia 1.10, the latency is now around 3 seconds. Better. But we still have room for improvement.

```
oscar@Oscars-MBP JuMP % julia
_
_ _ _(_)_ | Documentation: https://docs.julialang.org
(_) | (_) (_) |
_ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help.
| | | | | | |/ _` | |
| | |_| | | | (_| | | Version 1.10.0 (2023-12-25)
_/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |
julia> @time using JuMP, HiGHS
2.645880 seconds (1.17 M allocations: 82.527 MiB, 4.00% gc time, 0.85% compilation time)
julia> @time begin
model = Model(HiGHS.Optimizer)
set_silent(model)
@variable(model, x >= 0)
@variable(model, 0 <= y <= 3)
@objective(model, Min, 12x + 20y)
@constraint(model, c1, 6x + 8y >= 100)
@constraint(model, c2, 7x + 12y >= 120)
optimize!(model)
end
0.214786 seconds (241.91 k allocations: 16.071 MiB, 97.47% compilation time: 45% of which was recompilation)
```

## Future plans

We have big plans for this year. We will continue to improve JuMP’s nonlinear support, particularly as solvers like Gurobi and Xpress release and refine support for MINLPs. On the feature front, a priority is adding support for modeling with SI units. And finally, JuMP-dev 2024 will be held July 19-21 2024 in Montréal.

Here’s to another big year for JuMP!