Infeasible

Infeasible is an adjective used to describe a problem, system, or set of constraints that cannot be satisfied. A problem is infeasible when there is no assignment of values that meets all constraints; equivalently, the set of feasible solutions is empty.

In optimization and mathematical programming, infeasibility means the constraints of the formulation cannot be met simultaneously. This is distinct from unboundedness, where the objective can improve without bound while constraints hold. Infeasibility concerns the existence of any feasible point, not the value of the objective.

Solvers for linear programming, nonlinear programming, and constraint satisfaction may report infeasibility and, in many cases, provide an infeasibility certificate or proof that the constraint system cannot be satisfied. The existence of such certificates is supported by duality theorems and related results in optimization theory.

Examples include conflicting constraints such as x > 5 and x <= 5, or time-window constraints that demand incompatible resource usage. Infeasibility can arise in scheduling, network design, and constraint satisfaction problems when the constraints imply a contradiction.

Infeasibility is a common consideration across disciplines that use formal constraint systems. Detecting infeasibility informs reformulation, constraint relaxation, or the search for a consistent subset of constraints, helping to identify the source of the conflict and guide subsequent problem solving.