Basic Usage
First we load Convex itself, LinearAlgebra to access the identity matrix I
, and two solvers: SCS and GLPK.
using Convex
using LinearAlgebra
using SCS, GLPK
Linear program
\[\begin{array}{ll} \text{maximize} & c^T x \\ \text{subject to} & A x \leq b\\ & x \geq 1 \\ & x \leq 10 \\ & x_2 \leq 5 \\ & x_1 + x_4 - x_2 \leq 10 \\ \end{array}\]
x = Variable(4)
c = [1; 2; 3; 4]
A = I(4)
b = [10; 10; 10; 10]
constraints = [A * x <= b, x >= 1, x <= 10, x[2] <= 5, x[1] + x[4] - x[2] <= 10]
p = minimize(dot(c, x), constraints) # or c' * x
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 1 (4 scalar elements)
number of constraints : 5 (14 scalar elements)
number of coefficients : 36
number of atoms : 17
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 9.9998
Expression graph
minimize
└─ sum (affine; real)
└─ .* (affine; real)
├─ 4×1 Matrix{Int64}
└─ 4-element real variable (id: 705…155)
subject to
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ 4×1 Matrix{Int64}
├─ ≥ constraint (affine)
│ └─ + (affine; real)
│ ├─ 4-element real variable (id: 705…155)
│ └─ Convex.NegateAtom (constant; negative)
│ └─ …
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ 4-element real variable (id: 705…155)
│ └─ Convex.NegateAtom (constant; negative)
│ └─ …
⋮
We can also inspect the objective value and the values of the variables at the solution:
println(round(p.optval, digits = 2))
println(round.(evaluate(x), digits = 2))
println(evaluate(x[1] + x[4] - x[2]))
10.0
[1.0, 1.0, 1.0, 1.0]
0.9992362732533817
Matrix Variables and promotions
\[\begin{array}{ll} \text{minimize} & \| X \|_F + y \\ \text{subject to} & 2 X \leq 1\\ & X' + y \geq 1 \\ & X \geq 0 \\ & y \geq 0 \\ \end{array}\]
X = Variable(2, 2)
y = Variable()
# X is a 2 x 2 variable, and y is scalar. X' + y promotes y to a 2 x 2 variable before adding them
p = minimize(norm(X) + y, 2 * X <= 1, X' + y >= 1, X >= 0, y >= 0)
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 2 (5 scalar elements)
number of constraints : 4 (13 scalar elements)
number of coefficients : 25
number of atoms : 18
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 1.0002
Expression graph
minimize
└─ + (convex; real)
├─ norm2 (convex; positive)
│ └─ reshape (affine; real)
│ └─ …
└─ real variable (id: 673…980)
subject to
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ Convex.NegateAtom (constant; negative)
│ └─ …
├─ ≥ constraint (affine)
│ └─ + (affine; real)
│ ├─ reshape (affine; real)
│ │ └─ …
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ Convex.NegateAtom (constant; negative)
│ └─ …
├─ ≥ constraint (affine)
│ └─ + (affine; real)
│ ├─ 2×2 real variable (id: 321…265)
│ └─ Convex.NegateAtom (constant; negative)
│ └─ …
⋮
We can also inspect the values of the variables at the solution:
println(round.(evaluate(X), digits = 2))
println(evaluate(y))
p.optval
1.0002185951700004
Norm, exponential and geometric mean
\[\begin{array}{ll} \text{satisfy} & \| x \|_2 \leq 100 \\ & e^{x_1} \leq 5 \\ & x_2 \geq 7 \\ & \sqrt{x_3 x_4} \geq x_2 \end{array}\]
x = Variable(4)
p = satisfy(
norm(x) <= 100,
exp(x[1]) <= 5,
x[2] >= 7,
geomean(x[3], x[4]) >= x[2],
)
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 1 (4 scalar elements)
number of constraints : 4 (4 scalar elements)
number of coefficients : 3
number of atoms : 13
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
Expression graph
satisfy
└─ nothing
subject to
├─ ≤ constraint (convex)
│ └─ + (convex; real)
│ ├─ norm2 (convex; positive)
│ │ └─ …
│ └─ [-100;;]
├─ ≤ constraint (convex)
│ └─ + (convex; real)
│ ├─ exp (convex; positive)
│ │ └─ …
│ └─ [-5;;]
├─ ≥ constraint (affine)
│ └─ + (affine; real)
│ ├─ index (affine; real)
│ │ └─ …
│ └─ [-7;;]
⋮
PSD cone and Eigenvalues
y = Semidefinite(2)
p = maximize(eigmin(y), tr(y) <= 6)
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 1 (4 scalar elements)
number of constraints : 2 (5 scalar elements)
number of coefficients : 1
number of atoms : 4
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 3.0
Expression graph
maximize
└─ eigmin (concave; real)
└─ 2×2 real variable (id: 114…034)
subject to
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ sum (affine; real)
│ │ └─ …
│ └─ [-6;;]
├─ PSD constraint (convex)
│ └─ 2×2 real variable (id: 114…034)
└─ PSD constraint (convex)
└─ 2×2 real variable (id: 114…034)
x = Variable()
y = Variable((2, 2))
# PSD constraints
p = minimize(x + y[1, 1], y ⪰ 0, x >= 1, y[2, 1] == 1)
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 2 (5 scalar elements)
number of constraints : 3 (6 scalar elements)
number of coefficients : 2
number of atoms : 5
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 1.0009
Expression graph
minimize
└─ + (affine; real)
├─ real variable (id: 114…310)
└─ index (affine; real)
└─ 2×2 real variable (id: 118…881)
subject to
├─ PSD constraint (convex)
│ └─ 2×2 real variable (id: 118…881)
├─ ≥ constraint (affine)
│ └─ + (affine; real)
│ ├─ real variable (id: 114…310)
│ └─ [-1;;]
└─ == constraint (affine)
└─ + (affine; real)
├─ index (affine; real)
│ └─ …
└─ [-1;;]
Mixed integer program
\[\begin{array}{ll} \text{minimize} & \sum_{i=1}^n x_i \\ \text{subject to} & x \in \mathbb{Z}^n \\ & x \geq 0.5 \\ \end{array}\]
x = Variable(4, IntVar)
p = minimize(sum(x), x >= 0.5)
solve!(p, GLPK.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 1 (4 scalar elements)
number of constraints : 1 (4 scalar elements)
number of coefficients : 5
number of atoms : 4
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : NO_SOLUTION
objective value : 4.0
Expression graph
minimize
└─ sum (affine; real)
└─ 4-element real variable (id: 170…170)
subject to
└─ ≥ constraint (affine)
└─ + (affine; real)
├─ 4-element real variable (id: 170…170)
└─ Convex.NegateAtom (constant; negative)
└─ …
And the value of x
at the solution:
evaluate(x)
4-element Vector{Float64}:
1.0
1.0
1.0
1.0
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