Infeasibility certificates
When given a conic problem that is infeasible or unbounded, some solvers can produce a certificate of infeasibility. This page explains what a certificate of infeasibility is, and the related conventions that MathOptInterface adopts.
Conic duality
MathOptInterface uses conic duality to define infeasibility certificates. A full explanation is given in the section Duality, but here is a brief overview.
Minimization problems
For a minimization problem in geometric conic form, the primal is:
\[\begin{align} & \min_{x \in \mathbb{R}^n} & a_0^\top x + b_0 \\ & \;\;\text{s.t.} & A_i x + b_i & \in \mathcal{C}_i & i = 1 \ldots m, \end{align}\]
and the dual is a maximization problem in standard conic form:
\[\begin{align} & \max_{y_1, \ldots, y_m} & -\sum_{i=1}^m b_i^\top y_i + b_0 \\ & \;\;\text{s.t.} & a_0 - \sum_{i=1}^m A_i^\top y_i & = 0 \\ & & y_i & \in \mathcal{C}_i^* & i = 1 \ldots m, \end{align}\]
where each $\mathcal{C}_i$ is a closed convex cone and $\mathcal{C}_i^*$ is its dual cone.
Maximization problems
For a maximization problem in geometric conic form, the primal is:
\[\begin{align} & \max_{x \in \mathbb{R}^n} & a_0^\top x + b_0 \\ & \;\;\text{s.t.} & A_i x + b_i & \in \mathcal{C}_i & i = 1 \ldots m, \end{align}\]
and the dual is a minimization problem in standard conic form:
\[\begin{align} & \min_{y_1, \ldots, y_m} & \sum_{i=1}^m b_i^\top y_i + b_0 \\ & \;\;\text{s.t.} & a_0 + \sum_{i=1}^m A_i^\top y_i & = 0 \\ & & y_i & \in \mathcal{C}_i^* & i = 1 \ldots m. \end{align}\]
Unbounded problems
A problem is unbounded if and only if:
- there exists a feasible primal solution
- the dual is infeasible.
A feasible primal solution—if one exists—can be obtained by setting ObjectiveSense to FEASIBILITY_SENSE before optimizing. Therefore, most solvers stop after they prove the dual is infeasible via a certificate 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 $\eta > 0$:
\[A_i (x + \eta d) + b_i \in \mathcal{C}_i,\ \ i = 1 \ldots m,\]
and (for minimization problems):
\[a_0^\top (x + \eta d) + b_0 < a_0^\top x + b_0,\]
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 dual infeasibility:
TerminationStatusmust beDUAL_INFEASIBLEPrimalStatusmust beINFEASIBILITY_CERTIFICATEVariablePrimalmust be the corresponding value of $d$ConstraintPrimalmust be the corresponding value of $A_i d$ObjectiveValuemust be the value $a_0^\top d$. Note that this is the value of the objective function atd, ignoring the constantb_0.
The choice of whether to scale the ray $d$ to have magnitude 1 is left to the solver.
Infeasible problems
A certificate of primal infeasibility is an improving ray of the dual problem. However, because infeasibility is independent of the objective function, we first homogenize the primal problem by removing its objective.
For a minimization problem, a dual improving ray is some vector $d$ such that for all $\eta > 0$:
\[\begin{align} -\sum_{i=1}^m A_i^\top (y_i + \eta d_i) & = 0 \\ (y_i + \eta d_i) & \in \mathcal{C}_i^* & i = 1 \ldots m, \end{align}\]
and:
\[-\sum_{i=1}^m b_i^\top (y_i + \eta d_i) > -\sum_{i=1}^m b_i^\top y_i,\]
for any feasible dual solution $y$. The latter simplifies to $-\sum_{i=1}^m b_i^\top d_i > 0$. For a maximization problem, the inequality is $\sum_{i=1}^m b_i^\top d_i < 0$. (Note that these are the same inequality, modulo a - sign.)
If the solver has found a certificate of primal infeasibility:
TerminationStatusmust beINFEASIBLEDualStatusmust beINFEASIBILITY_CERTIFICATEConstraintDualmust be the corresponding value of $d$DualObjectiveValuemust be the value $-\sum_{i=1}^m b_i^\top d_i$ for minimization problems and $\sum_{i=1}^m b_i^\top d_i$ for maximization problems.
The choice of whether to scale the ray $d$ to have magnitude 1 is left to the solver.
Infeasibility certificates of variable bounds
Many linear solvers (for example, Gurobi) do not provide explicit access to the primal infeasibility certificate of a variable bound. However, given a set of linear constraints:
\[\begin{align} l_A \le A x \le u_A \\ l_x \le x \le u_x, \end{align}\]
the primal certificate of the variable bounds can be computed using the primal certificate associated with the affine constraints, $d$. (Note that $d$ will have one element for each row of the $A$ matrix, and that some or all of the elements in the vectors $l_A$ and $u_A$ may be $\pm \infty$. If both $l_A$ and $u_A$ are finite for some row, the corresponding element in `d must be 0.)
Given $d$, compute $\bar{d} = d^\top A$. If the bound is finite, a certificate for the lower variable bound of $x_i$ is $\max\{\bar{d}_i, 0\}$, and a certificate for the upper variable bound is $\min\{\bar{d}_i, 0\}$.