Which of the following statements about Slater’s condition is NOT true?
Slater’s condition implies the KKT conditions
Slater’s condition requires the optimal solution to satisfy all the inequality constraints strictly
Slater’s condition requires the inequality constraints to be convex
Slater’s condition is a sufficient condition for strong duality to hold
The correct answer and explanation is :
The correct answer is:
“Slater’s condition requires the inequality constraints to be convex.”
Explanation:
Slater’s condition is a crucial concept in optimization theory, particularly in the context of convex programming and duality. It plays a significant role in ensuring strong duality holds between the primal and dual problems. Let’s break down each of the statements:
- “Slater’s condition implies the KKT conditions”:
- This is true. Slater’s condition is a sufficient condition for the KKT (Karush-Kuhn-Tucker) conditions to hold. In optimization, Slater’s condition provides a way to guarantee that the KKT conditions are satisfied in convex optimization problems. If Slater’s condition is satisfied, it is enough to ensure that the KKT optimality conditions hold.
- “Slater’s condition requires the optimal solution to satisfy all the inequality constraints strictly”:
- This is true. Slater’s condition requires that for a convex optimization problem with inequality constraints, there exists at least one feasible point where all inequality constraints are strictly satisfied. This strict feasibility is essential for ensuring that the problem satisfies certain regularity conditions necessary for strong duality.
- “Slater’s condition requires the inequality constraints to be convex”:
- This statement is NOT true. While Slater’s condition is typically discussed in the context of convex problems, it does not require the inequality constraints themselves to be convex. It requires the problem to be convex, and the inequality constraints should be continuous and satisfy strict feasibility (i.e., there is an interior point where all inequalities are strict). However, the individual constraints do not have to be convex for Slater’s condition to hold.
- “Slater’s condition is a sufficient condition for strong duality to hold”:
- This is true. Slater’s condition is a well-known sufficient condition for strong duality in convex optimization problems. When Slater’s condition holds, it guarantees that the optimal value of the primal problem equals the optimal value of the dual problem, ensuring strong duality.
In summary, Slater’s condition does not impose a requirement on the convexity of the inequality constraints. The key requirement is that there must be a strictly feasible point that satisfies the constraints, which facilitates strong duality and the validity of KKT conditions.