What is the Kuhn-Tucker condition related to the Lagrange multipliers associated with equality constraints that are binding at the optimum? They are zero They are positive They are negative They are infinite
Answer (1)
The Kuhn-Tucker conditions, also known as the Karush-Kuhn-Tucker (KKT) conditions, are a set of mathematical criteria used to find the optimal solutions to certain types of optimization problems, particularly those involving inequality constraints.
In the context of the Lagrange multipliers related to equality constraints that are binding at the optimum, the Kuhn-Tucker condition specifies that these multipliers must be zero.
Here’s a step-by-step explanation of why this is the case:
Optimization Problem Setup :
Suppose you have an optimization problem where you want to maximize or minimize a function subject to some constraints.
This function can have both equality constraints g i ( x ) = 0 and inequality constraints h j ( x ) ≤ 0 .
Role of Lagrange Multipliers :
Lagrange multipliers are introduced to transform a constrained problem into a form that can be analyzed more easily.
For equality constraints, if a constraint is 'binding' at the optimum, it means that at the optimal point, the constraint is exactly satisfied (i.e., actually equals zero) and affects the optimality.
Kuhn-Tucker Conditions :
These include various prerequisites that must be met for a point to be considered an optimal solution.
For the equality constraints that are binding, the KKT conditions require that the corresponding Lagrange multipliers be zero. This is because the equality constraint is precisely holding 'tight' and does not allow any deviation from its set value at the optimum.
In summary, for equality constraints that are binding at the optimum, the associated Lagrange multipliers must indeed be zero. Thus, the answer to the question is: They are zero .
