Duality gap

Definition

The duality gap refers to the difference between the optimal values of a primal and its corresponding dual optimization problem. This gap can provide insights into the relationship between these two problems, indicating whether strong duality holds or if there are potential issues with feasibility or boundedness.

5 Must Know Facts For Your Next Test

1. A zero duality gap indicates that strong duality holds, which means that both primal and dual problems have the same optimal value.
2. In cases where the primal problem is feasible but the dual is infeasible, a non-zero duality gap will occur, suggesting potential issues in problem formulation.
3. The duality gap can serve as an indicator of how close a solution is to being optimal; smaller gaps imply closer approximations to the true optimal value.
4. For convex optimization problems, particularly those satisfying certain regularity conditions, the duality gap tends to be zero under appropriate circumstances.
5. Understanding the duality gap is crucial for optimization algorithms, as it helps determine convergence properties and guides algorithmic adjustments.

Review Questions

• How does the concept of duality gap relate to strong duality in optimization problems?
  • The duality gap directly connects to strong duality by indicating whether the optimal values of the primal and dual problems are equal. When the duality gap is zero, it means that strong duality holds, confirming that both problems yield the same optimal value. Conversely, a non-zero duality gap suggests that strong duality does not hold, which could indicate issues such as infeasibility or unboundedness in one of the problems.
• In what situations might you encounter a non-zero duality gap in optimization problems, and what implications does this have for the solutions?
  • A non-zero duality gap may occur when either the primal or the dual problem is infeasible, or when neither problem is bounded. This situation suggests that there are challenges with finding optimal solutions and highlights areas where adjustments may be needed. It indicates potential inconsistencies between the constraints or objectives of the primal and dual problems, impacting their respective solution spaces.
• Evaluate how interior point methods utilize information about the duality gap in their optimization process.
  • Interior point methods leverage information about the duality gap to improve convergence towards optimal solutions. By monitoring this gap throughout iterations, these methods can adjust their search direction and step sizes accordingly. A decreasing duality gap during iterations signifies progress toward optimality, allowing these algorithms to effectively navigate through feasible regions of both primal and dual spaces while ensuring they are closing in on optimal values.

Related terms

Primal Problem: The original optimization problem formulated to minimize or maximize a certain objective function subject to constraints.

Dual Problem: The optimization problem derived from the primal problem, typically focusing on maximizing or minimizing a different objective that relates to the constraints of the primal.

Strong Duality: A condition where the optimal values of the primal and dual problems are equal, meaning the duality gap is zero.