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> using Convex, SCS
julia> m = 4;  n = 55
julia> A = randn(m, n); b = randn(m)4-element Vector{Float64}:
 -0.5927610152342344
  0.5173544393784156
 -0.21310982928576944
 -0.72107968299556
julia> x = Variable(n)Variable
size: (5, 1)
sign: real
vexity: affine
id: 733…189
julia> problem = minimize(sumsquares(A * x - b), [x >= 0])Problem statistics
  problem is DCP         : true
  number of variables    : 1 (5 scalar elements)
  number of constraints  : 1 (5 scalar elements)
  number of coefficients : 31
  number of atoms        : 6

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

Expression graph
  minimize
   └─ qol (convex; positive)
      ├─ + (affine; real)
      │  ├─ * (affine; real)
      │  │  ├─ …
      │  │  └─ …
      │  └─ 4×1 Matrix{Float64}
      └─ [1;;]
  subject to
   └─ ≥ constraint (affine)
      └─ + (affine; real)
         ├─ 5-element real variable (id: 733…189)
         └─ Convex.NegateAtom (constant; negative)
            └─ …
julia> solve!(problem, SCS.Optimizer; silent = true)Problem statistics
  problem is DCP         : true
  number of variables    : 1 (5 scalar elements)
  number of constraints  : 1 (5 scalar elements)
  number of coefficients : 31
  number of atoms        : 6

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : FEASIBLE_POINT
  objective value    : 0.0

Expression graph
  minimize
   └─ qol (convex; positive)
      ├─ + (affine; real)
      │  ├─ * (affine; real)
      │  │  ├─ …
      │  │  └─ …
      │  └─ 4×1 Matrix{Float64}
      └─ [1;;]
  subject to
   └─ ≥ constraint (affine)
      └─ + (affine; real)
         ├─ 5-element real variable (id: 733…189)
         └─ Convex.NegateAtom (constant; negative)
            └─ …
julia> problem.statusOPTIMAL::TerminationStatusCode = 1
julia> problem.optval4.101528223800368e-7
julia> x.value5×1 Matrix{Float64}:
 0.41358470955079846
 0.245090958084477
 0.4111181215831069
 0.8509570025155792
 0.18183695552228096