Add more tests

docs/src/background/infeasibility_certificates.md

@@ -61,15 +61,16 @@ and the dual is a minimization problem in standard conic form:
 
 ## Unbounded models
 
-If a model is unbounded, then two things are true:
+If a model is unbounded, two things are true:
  1. there exists a feasible primal solution
  2. the dual is infeasible.
 
 A feasible primal solution---if one exists---can be obtained by setting
 [`ObjectiveSense`](@ref) to `FEASIBILITY_SENSE` and then optimizing. Therefore,
 most solvers terminate after they prove the dual is infeasible via a certificate
-of dual infeasibility. This is also the reason that MathOptInterface defines the
-`DUAL_INFEASIBLE` status instead of `UNBOUNDED`.
+of dual infeasibility, but _before_ they have found a feasible primal solution.
+This is also the reason that MathOptInterface defines the `DUAL_INFEASIBLE`
+status instead of `UNBOUNDED`.
 
 A certificate of dual infeasibility is an improving ray of the primal problem.
 That is, there exists some vector ``d`` such that for all ``\eta > 0``:
@@ -83,14 +84,18 @@ a_0^\top (x + \eta d) + b_0 < a_0^\top x + b_0,
 for any feasible point ``x``. The latter simplifies to ``a_0^\top d < 0``. For
 maximization problems, the inequality is reversed, so that ``a_0^\top d > 0``.
 
-If the solver has found a certificate of infeasibility:
+If the solver has found a certificate of dual infeasibility:
 
  * [`TerminationStatus`](@ref) must be `DUAL_INFEASIBLE`
  * [`PrimalStatus`](@ref) must be `INFEASIBILITY_CERTIFICATE`
  * [`VariablePrimal`](@ref) must be the corresponding value of ``d``
  * [`ObjectiveValue`](@ref) must be the value ``a_0^\top d``. Note that this is
    the value of the objective function at `d`, ignoring the constant `b_0`.
 
+!!! note
+    The choice of whether to scale the ray ``d`` to have magnitude `1` is left
+    to the solver.
+
 ## Infeasible models
 
 A certificate of primal infeasibility is an improving ray of the dual problem.
@@ -118,6 +123,10 @@ If the solver has found a certificate of infeasibility:
 
  * [`TerminationStatus`](@ref) must be `INFEASIBLE`
  * [`DualStatus`](@ref) must be `INFEASIBILITY_CERTIFICATE`
- * [`ConstraintPrimal`](@ref) must be the corresponding value of ``d``
+ * [`ConstraintDual`](@ref) must be the corresponding value of ``d``
  * [`DualObjectiveValue`](@ref) must be the value
    ``\sum_{i=1}^m b_i^\top d_i``.
+
+!!! note
+    The choice of whether to scale the ray ``d`` to have magnitude `1` is left
+    to the solver.