Active constraints

Definition

Active constraints are those restrictions in an optimization problem that are currently limiting the feasible solutions at the optimal point. When a solution is found, active constraints are the ones that hold with equality, meaning they directly influence the solution's position within the feasible region. Understanding which constraints are active is crucial for analyzing the solution's stability and helps in assessing the feasibility of potential changes to the problem.

5 Must Know Facts For Your Next Test

1. Active constraints define the boundary of the feasible region where the optimal solution lies, meaning they are critical to understanding optimization outcomes.
2. A constraint is considered active at a solution if it is met exactly (with equality), while inactive constraints can be satisfied without affecting the solution.
3. In problems with multiple constraints, it is possible for only some of them to be active at a given optimal solution, highlighting their selective influence.
4. The identification of active constraints can change if parameters of the problem are adjusted, which can lead to different optimal solutions.
5. Active constraints play a significant role in duality theory, as they help establish relationships between primal and dual solutions through concepts like complementary slackness.

Review Questions

• How do active constraints differ from inactive constraints in the context of optimization problems?
  • Active constraints differ from inactive constraints in that active constraints are satisfied with equality and directly limit the feasible solutions at the optimal point. In contrast, inactive constraints are satisfied with strict inequalities and do not affect the current solution. Understanding this distinction is vital because knowing which constraints are active can help identify how changes to the problem might affect the optimal solution.
• Discuss how identifying active constraints can impact the analysis of an optimization problem's sensitivity to changes.
  • Identifying active constraints is key when analyzing an optimization problem's sensitivity because these constraints define the exact limits within which solutions operate. If parameters change and a previously inactive constraint becomes active, or vice versa, it could shift the optimal solution and change feasibility conditions. Thus, knowing which constraints are active allows for better predictions about how variations will affect outcomes and decision-making in optimization.
• Evaluate the role of active constraints in establishing strong duality and complementary slackness in optimization theory.
  • Active constraints are essential for establishing strong duality because they directly relate primal and dual problems by indicating where both achieve optimal values. The concept of complementary slackness states that for each active constraint in the primal problem, there is a corresponding non-zero dual variable. This relationship emphasizes how changes in one formulation (primal) can impact its counterpart (dual) through these active boundaries, enhancing our understanding of solution dynamics and their implications in optimization scenarios.

Related terms

Inactive constraints: Constraints that do not affect the feasible solution at the optimal point because they are satisfied with strict inequality.

Feasible region: The set of all possible points that satisfy the constraints of an optimization problem.

KKT conditions: A set of necessary conditions for a solution in nonlinear programming to be optimal, which include the consideration of active constraints.