Bindinggrenzen

Bindinggrenzen (binding constraints) are constraints in an optimization problem that are active at the optimal solution, meaning they hold as equalities and restrict the feasible region at that point. They contrast with non-binding (slack) constraints, which are satisfied strictly with room to spare in the optimum.

An individual constraint i is binding at a solution x* if the slack b_i - a_i^T x* equals zero. If the slack is positive, the constraint is non-binding. The amount of slack can be described by a slack variable s_i = b_i - a_i^T x; binding occurs when s_i = 0.

Bindinggrenzen matter because they determine how changes to the problem data affect the optimum. In standard linear programming form, the system has dual (shadow) prices associated with constraints; a binding constraint typically carries a positive dual value, indicating a potential change in the objective when the constraint’s right-hand side b_i is altered. Non-binding constraints usually have zero dual value at the optimum and do not immediately affect the objective when perturbed.

Example: maximize z = 5x + 4y subject to x + y ≤ 6, 2x + y ≤ 8, x ≥ 0, y ≥ 0. The intersection of the two binding constraints x + y = 6 and 2x + y = 8 yields x = 2, y = 4, with z = 26. Here both constraints are binding at the optimum, and their right-hand sides directly influence the optimal value via the dual variables.

Bindinggrenzen are a central concept in sensitivity analysis and are fundamental to understanding how optimal solutions respond to changes in the problem data.