Document how to use MOI.VectorAffineFunction

docs/src/optimization.md

@@ -126,3 +126,66 @@ m = Model()
 
 poly = polyhedron(m, CDDLib.Library(:exact))
 ```
+
+## Using a Polyhedra Optimizer with MathOptInterface
+
+Polyhedra Optimizers by dafault support only a few constraint types in MathOptInterface (MOI).
+Apply `MOI.Bridges.full_bridge_optimizer` to a Polyhedra Optimizer to enable a broader set of constraint types, such as `VectorAffineFunction`:
+see the [list](http://www.juliaopt.org/MathOptInterface.jl/dev/apimanual/#Constraints-by-function-set-pairs-1) from MOI.
+
+As an example, consider the linear program:
+```math
+\[
+\max\ c x \quad \text{s.t.}\ A x \leq b,\ x \geq 0
+\]
+```
+where
+```julia
+A = [1 2; 3 1]
+b = [1, 2]
+c = [1, 1]
+```
+
+Let us solve this program with `CDDLib.Optimizer` in exact arithmetic.
+To set up:
+
+```julia
+using CDDLib
+using MathOptInterface
+const MOI = MathOptInterface
+
+m, n = size(A)
+T = Rational{BigInt}
+
+# Enable `VectorAffineTerm`, `VectorOfVariables`, `Nonnegatives`, `Nonpositives`
+optimizer = MOI.Bridges.full_bridge_optimizer(CDDLib.Optimizer{T}(), T)
+
+# max cx
+x = MOI.add_variables(optimizer, n)
+MOI.set(optimizer, MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
+        MOI.ScalarAffineFunction{T}(MOI.ScalarAffineTerm{T}.(c, x), 0))
+MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+
+# Ax - b <= 0
+terms = MOI.VectorAffineTerm{T}.(
+    1:m, MOI.ScalarAffineTerm{T}.(A, reshape(x, 1, n))
+)
+f = MOI.VectorAffineFunction{T}(vec(terms), -b)
+MOI.add_constraint(optimizer, f, MOI.Nonpositives(m))
+
+# x >= 0
+MOI.add_constraint(optimizer, MOI.VectorOfVariables(x), MOI.Nonnegatives(n))
+```
+
+To solve:
+
+```julia
+MOI.optimize!(optimizer)
+MOI.get(optimizer, MOI.VariablePrimal(), x)
+```
+
+```
+2-element Array{Rational{BigInt},1}:
+ 3//5
+ 1//5
+```