README.md

SimplePDHG.jl

I wrote this simple educational Julia package (less than 350 lines of code) for the MIT Course 15.084/6.7220 Nonlinear Optimization. The purpose of this package is to demonstrate to the students how simple it is to implement an optimization algorithm in Julia and connect it to the optimization modeling language JuMP so that anyone can use your algorithm.

Big thanks to Oscar Dowson for providing MathOptInterface.jl code to connect this simple solver to JuMP! (discourse link)

What does SimplePDHG.jl do?

This is an educational package used to demonstrate the ease of implementing an algorithm in Julia and incorporating it with one of Julia's main optimization modeling language JuMP. The package is designed to solve linear programming problems of the form:

minimize    c'x
subject to  A x = b
            G x ≤ h
            x ∈ ℝ^n

where x is the decision variable. Under the hood the SimplePDHG.jl implements the vanilla PDHG algorithm (see Section 3.3 of this book) to solve standard form linear optimization problem of the form min{c'x ∣ Ax=b, x ≥ 0, x ∈ ℝ^n}.

Installation

Type the following in Julia REPL to the stable version:

] add SimplePDHG

To get the latest branch, type:

] add https://github.com/Shuvomoy/SimplePDHG.jl.git

Usage through JuMP

using JuMP, SimplePDHG
model =  Model(SimplePDHG.Optimizer)
@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)
println("Objective value: ", objective_value(model))
println("x = ", value(x))
println("y = ", value(y))

Output should be:

Objective value: 205.000090068938
x = 14.999887019427522
y = 1.2500722917903861

Vector syntax in JuMP

Thanks to JuMP and MathOptInterface.jl , we can use vectorized syntax to solve our optimization problem as well.

# data 
A = [1 1 9 5; 3 5 0 8; 2 0 6 13]
b = [7, 3, 5]
c = [1, 3, 5, 2]
m, n = size(A)
G = [0.5012005468024234 -1.5806753104910911 1.1908183108070869 1.6527613262371468; -1.7596263752677483 -0.5235246034519885 0.4618550523688477 0.4871842582808355; -0.6305269735894394 0.023788955821653315 -0.5208935392017503 -1.667410808905106; 1.02249016425841 0.6890017766482583 1.2904648745012357 1.398062622113161; -0.9763001854265912 0.866180139889124 -0.18426778358700338 1.1436405988912726; 0.4004591856282607 -0.6315453522080423 -0.32707956849441 -1.192277331736516];
h = 2*ones(2*m)

# JuMP code
using JuMP, SimplePDHG
model =  Model(SimplePDHG.Optimizer)
@variable(model, x[1:n] >= 0)
@objective(model, Min, c'*x)
@constraint(model, A*x .== b)
@constraint(model, G*x .<= h)
optimize!(model)
println("Objective value: ", objective_value(model))
println("x = ", value.(x))
x_star = value.(x)

The output should be:

Objective value: 4.922528390226832

x = [0.42344643304517904, 0.34592985413549193, 0.6922584789550353, 0.0]