# Quick Tutorial

Consider a constrained least squares problem

\begin{aligned} \begin{array}{ll} \text{minimize} & \|Ax - b\|_2^2 \\ \text{subject to} & x \geq 0 \end{array} \end{aligned}

with variable $x\in \mathbf{R}^{n}$, and problem data $A \in \mathbf{R}^{m \times n}$, $b \in \mathbf{R}^{m}$.

This problem can be solved in Convex.jl as follows: :

julia> # Make the Convex.jl module available
using Convex, SCS

julia> # Generate random problem data
m = 4;  n = 5
5

julia> A = randn(m, n); b = randn(m, 1)
4×1 Array{Float64,2}:
-0.5982561974146283
1.8285635154246862
0.7473287766124718
1.9114497945229456

julia> # Create a (column vector) variable of size n x 1.
x = Variable(n)
Variable
size: (5, 1)
sign: real
vexity: affine
id: 122…600

julia> # The problem is to minimize ||Ax - b||^2 subject to x >= 0
# This can be done by: minimize(objective, constraints)
problem = minimize(sumsquares(A * x - b), [x >= 0])
minimize
└─ qol_elem (convex; positive)
├─ norm2 (convex; positive)
│  └─ + (affine; real)
│     ├─ …
│     └─ …
└─ [1.0]
subject to
└─ >= constraint (affine)
├─ 5-element real variable (id: 122…600)
└─ 0

status: solve! not called yet

julia> # Solve the problem by calling solve!
solve!(problem, () -> SCS.Optimizer(verbose=false))

julia> # Check the status of the problem
problem.status # :Optimal, :Infeasible, :Unbounded etc.
OPTIMAL::TerminationStatusCode = 1

julia> # Get the optimum value
problem.optval
7.791884997857074