All of the examples can be found in Jupyter notebook form here.
Convex Optimization in Julia
Madeleine Udell | ISMP 2015
Convex.jl team
- Convex.jl: Madeleine Udell, Karanveer Mohan, David Zeng, Jenny Hong
Collaborators/Inspiration:
- CVX: Michael Grant, Stephen Boyd
- CVXPY: Steven Diamond, Eric Chu, Stephen Boyd
- JuliaOpt: Miles Lubin, Iain Dunning, Joey Huchette
# initial package installation
# Make the Convex.jl module available
using Convex, SparseArrays, LinearAlgebra
using SCS # first order splitting conic solver [O'Donoghue et al., 2014]
# Generate random problem data
m = 50;
n = 100;
A = randn(m, n)
x♮ = sprand(n, 1, 0.5) # true (sparse nonnegative) parameter vector
noise = 0.1 * randn(m) # gaussian noise
b = A * x♮ + noise # noisy linear observations
# Create a (column vector) variable of size n.
x = Variable(n)
# nonnegative elastic net with regularization
λ = 1
μ = 1
problem = minimize(
square(norm(A * x - b)) + λ * square(norm(x)) + μ * norm(x, 1),
x >= 0,
)
# Solve the problem by calling solve!
solve!(problem, SCS.Optimizer; silent_solver = true)
println("problem status is ", problem.status) # :Optimal, :Infeasible, :Unbounded etc.
println("optimal value is ", problem.optval)
problem status is OPTIMAL
optimal value is 32.50981330008917
using Interact, Plots
# Interact.WebIO.install_jupyter_nbextension() # might be helpful if you see `WebIO` warnings in Jupyter
@manipulate throttle = 0.1 for λ in 0:0.1:5, μ in 0:0.1:5
global A
problem = minimize(
square(norm(A * x - b)) + λ * square(norm(x)) + μ * norm(x, 1),
x >= 0,
)
solve!(problem, SCS.Optimizer; silent_solver = true)
histogram(evaluate(x), xlims = (0, 3.5), label = "x")
end
Quick convex prototyping
Variables
# Scalar variable
x = Variable()
Variable
size: (1, 1)
sign: real
vexity: affine
id: 104…827
# (Column) vector variable
y = Variable(4)
Variable
size: (4, 1)
sign: real
vexity: affine
id: 665…780
# Matrix variable
Z = Variable(4, 4)
Variable
size: (4, 4)
sign: real
vexity: affine
id: 360…354
Expressions
Convex.jl allows you to use a wide variety of functions on variables and on expressions to form new expressions.
x + 2x
+ (affine; real)
├─ real variable (id: 104…827)
└─ * (affine; real)
├─ 2
└─ real variable (id: 104…827)
e = y[1] + logdet(Z) + sqrt(x) + minimum(y)
+ (concave; real)
├─ index (affine; real)
│ └─ 4-element real variable (id: 665…780)
├─ logdet (concave; real)
│ └─ 4×4 real variable (id: 360…354)
├─ geomean (concave; positive)
│ ├─ real variable (id: 104…827)
│ └─ [1.0]
└─ minimum (concave; real)
└─ 4-element real variable (id: 665…780)
Examine the expression tree
e.children[2]
logdet (concave; real)
└─ 4×4 real variable (id: 360…354)
Constraints
A constraint is convex if convex combinations of feasible points are also feasible. Equivalently, feasible sets are convex sets.
In other words, convex constraints are of the form
convexExpr <= 0
concaveExpr >= 0
affineExpr == 0
x <= 0
<= constraint (affine)
├─ real variable (id: 104…827)
└─ 0
square(x) <= sum(y)
<= constraint (convex)
├─ qol_elem (convex; positive)
│ ├─ real variable (id: 104…827)
│ └─ [1.0]
└─ sum (affine; real)
└─ 4-element real variable (id: 665…780)
M = Z
for i in 1:length(y)
global M += rand(size(Z)...) * y[i]
end
M ⪰ 0
sdp constraint (affine)
└─ + (affine; real)
├─ 4×4 real variable (id: 360…354)
├─ * (affine; real)
│ ├─ 4×4 Matrix{Float64}
│ └─ index (affine; real)
│ └─ …
├─ * (affine; real)
│ ├─ 4×4 Matrix{Float64}
│ └─ index (affine; real)
│ └─ …
├─ * (affine; real)
│ ├─ 4×4 Matrix{Float64}
│ └─ index (affine; real)
│ └─ …
└─ * (affine; real)
├─ 4×4 Matrix{Float64}
└─ index (affine; real)
└─ …
Problems
x = Variable()
y = Variable(4)
objective = 2 * x + 1 - sqrt(sum(y))
constraint = x >= maximum(y)
p = minimize(objective, constraint)
minimize
└─ + (convex; real)
├─ * (affine; real)
│ ├─ 2
│ └─ real variable (id: 181…453)
├─ 1
└─ - (convex; negative)
└─ geomean (concave; positive)
├─ …
└─ …
subject to
└─ >= constraint (convex)
├─ real variable (id: 181…453)
└─ maximum (convex; real)
└─ 4-element real variable (id: 375…886)
status: `solve!` not called yet
# solve the problem
solve!(p, SCS.Optimizer; silent_solver = true)
p.status
OPTIMAL::TerminationStatusCode = 1
evaluate(x)
0.24993992301833423
# can evaluate expressions directly
evaluate(objective)
0.49994748380953213
Pass to solver
call a MathProgBase
solver suited for your problem class
- see the list of Convex.jl operations to find which cones you're using
- see the list of solvers for an up-to-date list of solvers and which cones they support
to solve problem using a different solver, just import the solver package and pass the solver to the solve!
method: eg
using Mosek
solve!(p, Mosek.Optimizer)
Warmstart
# Generate random problem data
m = 50;
n = 100;
A = randn(m, n)
x♮ = sprand(n, 1, 0.5) # true (sparse nonnegative) parameter vector
noise = 0.1 * randn(m) # gaussian noise
b = A * x♮ + noise # noisy linear observations
# Create a (column vector) variable of size n.
x = Variable(n)
# nonnegative elastic net with regularization
λ = 1
μ = 1
problem = minimize(
square(norm(A * x - b)) + λ * square(norm(x)) + μ * norm(x, 1),
x >= 0,
)
@time solve!(problem, SCS.Optimizer; silent_solver = true)
λ = 1.5
@time solve!(problem, SCS.Optimizer; silent_solver = true, warmstart = true)
0.027181 seconds (8.36 k allocations: 1.540 MiB)
0.784736 seconds (883.60 k allocations: 51.032 MiB, 96.82% compilation time)
DCP examples
# affine
x = Variable(4)
y = Variable(2)
sum(x) + y[2]
+ (affine; real)
├─ sum (affine; real)
│ └─ 4-element real variable (id: 119…546)
└─ index (affine; real)
└─ 2-element real variable (id: 155…331)
2 * maximum(x) + 4 * sum(y) - sqrt(y[1] + x[1]) - 7 * minimum(x[2:4])
+ (convex; real)
├─ * (convex; real)
│ ├─ 2
│ └─ maximum (convex; real)
│ └─ 4-element real variable (id: 119…546)
├─ * (affine; real)
│ ├─ 4
│ └─ sum (affine; real)
│ └─ 2-element real variable (id: 155…331)
├─ - (convex; negative)
│ └─ geomean (concave; positive)
│ ├─ + (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ [1.0]
└─ - (convex; real)
└─ * (concave; real)
├─ 7
└─ minimum (concave; real)
└─ …
# not dcp compliant
log(x) + square(x)
+ (Convex.NotDcp; real)
├─ log (concave; real)
│ └─ 4-element real variable (id: 119…546)
└─ qol_elem (convex; positive)
├─ 4-element real variable (id: 119…546)
└─ 4×1 Matrix{Float64}
# $f$ is convex increasing and $g$ is convex
square(pos(x))
qol_elem (convex; positive)
├─ max (convex; positive)
│ ├─ 4-element real variable (id: 119…546)
│ └─ 0
└─ 4×1 Matrix{Float64}
# $f$ is convex decreasing and $g$ is concave
invpos(sqrt(x))
qol_elem (convex; positive)
├─ 4×1 Matrix{Float64}
└─ geomean (concave; positive)
├─ 4-element real variable (id: 119…546)
└─ 4×1 Matrix{Float64}
# $f$ is concave increasing and $g$ is concave
sqrt(sqrt(x))
geomean (concave; positive)
├─ geomean (concave; positive)
│ ├─ 4-element real variable (id: 119…546)
│ └─ 4×1 Matrix{Float64}
└─ 4×1 Matrix{Float64}
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