Active Constraint
In constrained optimization, an active constraint (or binding constraint) is a constraint that holds with equality at a candidate solution point. For inequality constraints of the form g(x) ≤ 0, a constraint is active at x* if g(x*) = 0. For equality constraints h(x) = 0, all constraints are active by definition at any feasible point. The set of active constraints at a point determines the local feasible geometry and is the primary object of analysis in optimality conditions.
The concept of active constraints is the bridge between the abstract geometry of feasible sets and the algebraic machinery of Lagrange multipliers. The KKT conditions apply only to the active set: inactive inequality constraints are ignored in the first-order optimality analysis, because they do not locally restrict the feasible region. This means that identifying the active set is equivalent to identifying the local shape of the problem.
Active set identification and the combinatorial problem
In nonlinear programming with m inequality constraints, there are 2^m possible active sets. The combinatorial explosion makes explicit enumeration impossible for large problems, which is why optimization algorithms use iterative methods to identify the active set implicitly. Interior-point methods approach the solution from the interior of the feasible region, never explicitly partitioning constraints into active and inactive; the active set emerges organically as the iterates approach the boundary. Active-set methods, by contrast, maintain an explicit working set of constraints believed to be active and update this set based on violation information.
The identification of the active set is not merely a computational convenience — it is a theoretical problem. For non-convex problems, a constraint may be active at a local minimum but inactive at the global minimum, and the algorithm's choice of which constraints to treat as active determines which minimum it converges to. This is one reason why global optimization is hard: the active set is not a local property but a global one, and local algorithms cannot determine it without exploring the entire feasible region.
Active constraints and sensitivity analysis
The active set at a solution determines the sensitivity of the optimal value to parameter changes. If a constraint is inactive, a small perturbation of the corresponding parameter does not affect the solution — the constraint is 'not felt' by the optimizer. If a constraint is active, the optimal value changes at a rate given by the corresponding Lagrange multiplier. This is the shadow price interpretation: the multiplier measures the marginal cost of the constraint, but only if the constraint is active.
In economic applications, this means that only binding constraints matter for policy. A resource constraint that is not binding (the economy is not using all available resources) has zero shadow price, and relaxing it further does not improve welfare. A binding constraint (the economy is at capacity) has a positive shadow price, and policy should target it. This is why active set identification is central to economic planning: the active constraints tell you where the bottlenecks are.
Active constraints in control and design
In control theory, active constraints often represent physical limits: actuator saturation, state bounds, safety envelopes. When a controller is operating with no active constraints, it behaves like an unconstrained controller. When constraints become active, the controller must switch to a different mode of operation, and the transition between unconstrained and constrained behavior is a source of discontinuity and nonlinearity. This is why model predictive control (MPC) is powerful: it explicitly accounts for active constraints in the optimization problem solved at each time step, and the optimizer naturally handles the transition between active and inactive regimes.
In engineering design, active constraints are typically the design requirements that are 'tight' — the stress limit, the weight limit, the cost limit. The inactive constraints have margin, which is good for robustness but indicates overdesign. The active constraints tell the designer which requirements are driving the design. A good design problem has a non-empty active set: if no constraints are active, the objective can be improved further, and the design is not optimal.