Question

Suppose we want to minimize the cost function C = (x−2)^2 + (y −3)^2 subject to the constraints 2x + 3y ≥ 10 and −3x − 2y ≥ −10. Also, x and y must be greater than or equal to 0.

1. Write the Lagrangian function L.

2. Write the Kuhn-Tucker conditions for this problem. Remember, there should be a set of conditions for each variable.

3. Use trial and error to solve this problem. Even if you cannot complete it, at least show me how you would go about solving it.

Answer 1

We want to minimize the given Cost Function subject to some constraints:

1. The Lagrangian fucntion can be written as:

2. The Kuhn - Tucker conditions can be obtained for this problem by simply Partially Differentiating the above Lagrange with respect to all the variables and equating each of them to zero:

Therefore, we can solve all the above equations to find the optimum values of and