Problem 3. QP. Consider the following quadratic programming model: Max Z- 20x1 +

Question

Problem 3. QP. Consider the following quadratic programming model: Max Z- 20x1 +40X2 9x12- 6x22 + 6X1X2 X1 + X2 X2 le 5 le 3 a) Derive the Kuhn-Tucker conditions for this problem b) Consider the point: X1-2 X2 3 Based on the Kuhn-Tucker conditions, could this point be the optimal point? Explain why or why not. c) Bonus. Excel Solver will easily run a QP problem. Set it up as you would an LP the objective function to fit the Quadratic form. Solve this problem by computer software for 5 bonus homework points. , but define

Explanation / Answer

Answer:

Kuhn Tucker Conditions:

It is both a necessary and sufficient conditions if the objective function is concave and each constraint is linear.

a. Lagrange function is : 20X1 + 40X2 - 9X12 - 6X22 + 6X1X2 + g1 ( 5 - X1 - X2 ) + g2 ( 3- X2 ) and the Kuhn -Tucker first - order conditions becomes:

Lx1  = 20 - 18X1 + 6X2 - g1 = 0 ( X1 is greater than equal to zero )

Lx2 = 40 - 12X2 + 6X1 - g1 - g2 = 0 ( X2 is greater than equal to zero )

L g1 = 5 - X1 - X2 = 0 ( g1 is greater than equal to zero )

L g2 = 3 - X2 = 0 ( g2 is greater than equal to zero )

Now, we have four equations and four unknown parameters as X1 , X2 , g1 and g2 .

Solving equation 4 , we get X2 = 3 .

Solving equation 3 and putting value of X2 = 3 , we get X1 = 2 .

Solving equation 1 with putting values of X1 and X2 , we get  g 1 = 2 .

Solving equation 2 we get  g2 = 14.  

b. Yes, these are the optimal points as function Z is maximised. Putting all these four values in the equation

Z = 20 * 2 + 40 * 3 - 9 * 4 - 6 * 9 + 6 * 2 * 3 + 2 * ( 5-2-3 ) + 14 * ( 3-3 )

Z = 106 Ans.

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