Active Constraints

Optimization · Axiom Academy

LESSON Active Constraints in Optimization Understanding which constraints truly matter at the optimal solution An active constraint (also called a binding constraint ) is one where the constraint is satisfied exactly at the optimal solution. For inequality constraints of the form g(x) ≤ 0, this means: At the optimal point x*, the constraint is "tight" - it's satisfied with equality. The constraint boundary passes directly through the optimal point. An inactive constraint is one where the constraint is satisfied with strict inequality at the optimal solution. For constraints g(x) ≤ 0, this means: At the optimal point x*, the constraint has "slack" - it's satisfied but not binding. The constraint boundary does not touch the optimal point. The active set at a point x* is the collection of all constraints that are active (binding) at that point: The active set tells us which constraints actually matter at the optimum. Only active constraints affect the KKT conditions and the geometry of the solution. 4. Example with Multiple Constraints Consider a 2D optimization problem with three inequality constraints. Watch how the optimal point is determined by which constraints are active: 5. Why Active Constraints Matter Understanding active constraints is essential because: Optimality conditions: The KKT multipliers are non-zero only for active constraints Algorithm efficiency: Many optimization algorithms identify the active set

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