# Convex Optimization in Julia

## 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
50×1 Matrix{Float64}:
-6.021565529578327
0.4862497893470268
1.1660980056235264
4.490736756364103
5.251290917775126
-4.5714554579907
-3.002120988244739
-2.2829575237564588
3.21402710163009
2.9665701034562697
⋮
6.883109824569101
0.5011905912894424
1.2741470859779214
2.4085866355099887
-1.610486215207384
1.7996228043647744
-4.758465080400447
-0.7828865793142671
-3.0420623369536117

Create a (column vector) variable of size n.

x = Variable(n)
Variable
size: (100, 1)
sign: real
vexity: affine
id: 268…099

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,
)
Problem statistics
problem is DCP         : true
number of variables    : 1 (100 scalar elements)
number of constraints  : 1 (100 scalar elements)
number of coefficients : 5_155
number of atoms        : 14

Solution summary
termination status : OPTIMIZE_NOT_CALLED
primal status      : NO_SOLUTION
dual status        : NO_SOLUTION

Expression graph
minimize
└─ + (convex; positive)
├─ qol_elem (convex; positive)
│  ├─ norm2 (convex; positive)
│  │  └─ …
│  └─ [1.0;;]
├─ * (convex; positive)
│  ├─ [1;;]
│  └─ qol_elem (convex; positive)
│     ├─ …
│     └─ …
└─ * (convex; positive)
├─ [1;;]
└─ sum (convex; positive)
└─ …
subject to
└─ ≥ constraint (affine)
└─ + (affine; real)
├─ 100-element real variable (id: 268…099)
└─ Convex.NegateAtom (constant; negative)
└─ …


Solve the problem by calling solve!

solve!(problem, SCS.Optimizer; silent = true)

println("problem status is ", problem.status)
println("optimal value is ", problem.optval)
problem status is OPTIMAL
optimal value is 25.689290176242185
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 = 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: 111…850

(Column) vector variable

y = Variable(4)
Variable
size: (4, 1)
sign: real
vexity: affine
id: 145…520

Matrix variable

Z = Variable(4, 4)
Variable
size: (4, 4)
sign: real
vexity: affine
id: 183…934

# 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: 111…850)
└─ * (affine; real)
├─ [2;;]
└─ real variable (id: 111…850)
e = y[1] + logdet(Z) + sqrt(x) + minimum(y)
+ (concave; real)
├─ index (affine; real)
│  └─ 4-element real variable (id: 145…520)
├─ logdet (concave; real)
│  └─ 4×4 real variable (id: 183…934)
├─ geomean (concave; positive)
│  ├─ real variable (id: 111…850)
│  └─ [1.0;;]
⋮

### Examine the expression tree

e.children[2]
logdet (concave; real)
└─ 4×4 real variable (id: 183…934)

# 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)
└─ + (affine; real)
├─ real variable (id: 111…850)
└─ [0;;]
square(x) <= sum(y)
≤ constraint (convex)
└─ + (convex; real)
├─ qol_elem (convex; positive)
│  ├─ real variable (id: 111…850)
│  └─ [1.0;;]
└─ Convex.NegateAtom (affine; real)
└─ sum (affine; real)
└─ …
M = Z
for i in 1:length(y)
global M += rand(size(Z)...) * y[i]
end
M ⪰ 0
PSD constraint (convex)
└─ + (affine; real)
├─ 4×4 real variable (id: 183…934)
├─ * (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)
Problem statistics
problem is DCP         : true
number of variables    : 2 (5 scalar elements)
number of constraints  : 1 (1 scalar elements)
number of coefficients : 3
number of atoms        : 8

Solution summary
termination status : OPTIMIZE_NOT_CALLED
primal status      : NO_SOLUTION
dual status        : NO_SOLUTION

Expression graph
minimize
└─ + (convex; real)
├─ * (affine; real)
│  ├─ [2;;]
│  └─ real variable (id: 116…971)
├─ [1;;]
└─ Convex.NegateAtom (convex; negative)
└─ geomean (concave; positive)
├─ …
└─ …
subject to
└─ ≥ constraint (convex)
└─ + (concave; real)
├─ real variable (id: 116…971)
└─ Convex.NegateAtom (concave; real)
└─ …


Solve the problem:

solve!(p, SCS.Optimizer; silent = true)
p.status
OPTIMAL::TerminationStatusCode = 1
evaluate(x)
0.2499992372452304

Can evaluate expressions directly:

evaluate(objective)
0.4999989328331398

## Pass to solver

call a MathProgBase solver suited for your problem class

to solve problem using a different solver, just import the solver package and pass the solver to the solve! method:

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
50×1 Matrix{Float64}:
-1.0813938133106962
4.0264046756355345
-4.039348933441501
-5.279272338197081
-4.616032868559761
5.306945990702305
-2.883570886421047
3.475336050338825
7.140299677062581
2.9045225655784774
⋮
2.4118139170692223
0.7628503149946742
2.172206721398036
-0.06148079589661848
2.057471254928378
-1.5832099506496735
-0.8160964195043132
0.934467104884041
6.959327482987776

Create a (column vector) variable of size n.

x = Variable(n)
Variable
size: (100, 1)
sign: real
vexity: affine
id: 167…601

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 = true)
λ = 1.5
@time solve!(problem, SCS.Optimizer; silent = true)#, warmstart = true) # FIXME
Problem statistics
problem is DCP         : true
number of variables    : 1 (100 scalar elements)
number of constraints  : 1 (100 scalar elements)
number of coefficients : 5_155
number of atoms        : 14

Solution summary
termination status : OPTIMAL
primal status      : FEASIBLE_POINT
dual status        : FEASIBLE_POINT
objective value    : 24.8818

Expression graph
minimize
└─ + (convex; positive)
├─ qol_elem (convex; positive)
│  ├─ norm2 (convex; positive)
│  │  └─ …
│  └─ [1.0;;]
├─ * (convex; positive)
│  ├─ [1;;]
│  └─ qol_elem (convex; positive)
│     ├─ …
│     └─ …
└─ * (convex; positive)
├─ [1;;]
└─ sum (convex; positive)
└─ …
subject to
└─ ≥ constraint (affine)
└─ + (affine; real)
├─ 100-element real variable (id: 167…601)
└─ Convex.NegateAtom (constant; negative)
└─ …


# DCP examples

• affine
x = Variable(4)
y = Variable(2)
sum(x) + y[2]
+ (affine; real)
├─ sum (affine; real)
│  └─ 4-element real variable (id: 166…465)
└─ index (affine; real)
└─ 2-element real variable (id: 143…863)
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: 166…465)
├─ * (affine; real)
│  ├─ [4;;]
│  └─ sum (affine; real)
│     └─ 2-element real variable (id: 143…863)
├─ Convex.NegateAtom (convex; negative)
│  └─ geomean (concave; positive)
│     ├─ + (affine; real)
│     │  ├─ …
│     │  └─ …
│     └─ [1.0;;]
⋮
• not DCP compliant
log(x) + square(x)
+ (Convex.NotDcp; real)
├─ log (concave; real)
│  └─ 4-element real variable (id: 166…465)
└─ qol_elem (convex; positive)
├─ 4-element real variable (id: 166…465)
└─ 4×1 Matrix{Float64}
• Composition $f\circ g$ where $f$ is convex increasing and $g$ is convex
square(pos(x))
qol_elem (convex; positive)
├─ max (convex; positive)
│  ├─ 4-element real variable (id: 166…465)
│  └─ [0;;]
└─ 4×1 Matrix{Float64}
• Composition $f\circ g$ where $f$ 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: 166…465)
└─ 4×1 Matrix{Float64}
• Composition $f\circ g$ where $f$ is concave increasing and $g$ is concave
sqrt(sqrt(x))
geomean (concave; positive)
├─ geomean (concave; positive)
│  ├─ 4-element real variable (id: 166…465)
│  └─ 4×1 Matrix{Float64}
└─ 4×1 Matrix{Float64}