Answer to Question #271086 in Operations Research for Tarurendra Kushwah

Question #271086

Write the Kuhn-Tucker conditions for the following problems and obtain the optimal solution:

Minimize                    Z= 2x1+3x2-x12- 2x22

Subject to      x1+3x2 ≤ 6,

 5x1+2x2 ≤ 10,

               x1, x2≥ 0


1
Expert's answer
2022-01-14T05:18:02-0500

1) gi(x)big_{i}(x^*)-b_{i} is feasible for i=1,2....mi=1,2....m


Where gi(x)g_{i}(x^*) is optional value


Writing the equation in residue form


f)x)=2x1+3x2x122x22f)x)=2x_{1}+3x_{2}-x_{1}^2-2x_{2}^{2}


g1(x)=x1+3x26g_1(x)=x_1+3x_2-6


g2(x)=5x1+2x210g_2(x)=5x_1+2x_2-10



Applying the Kuhn-Tucker conditions (Optimality Condition)


2)


(fx1fx2)\begin{pmatrix} \frac { \partial\>f}{\partial\>x_1} \\\\ \frac{\partial\>f}{\partial\>x_2} \end{pmatrix} +λ1+\lambda1 (g1x1g1x2)\begin{pmatrix} \frac{\partial\>g_1}{\partial\>x_1} \\\\ \frac{\partial\>g_1}{\partial\>x_2} \end{pmatrix} +λ2+\lambda2 (g2x1g2x2)=0\begin{pmatrix} \frac{\partial\>g_2}{\partial\>x_1} \\\\ \frac{\partial\>g_2}{\partial\>x_2} \end{pmatrix}=0



22x1+λ1+5λ2=0......(1)2-2x_1+\lambda_1+5\lambda_2=0......(1)

34x2+3λ1+2λ2=0.....(2)3-4x_2+3\lambda_1+2\lambda_2=0.....(2)



Feasibility condition


x1+3x26x_1+3x_2\le6 ......(3)......(3)

5x1+2x210  ......(4)5x_1+2x_2\le10\>\>......(4)


Complimentary slackness property


λ1(x1+3x26)=0\lambda_1(x_1+3x_2-6)=0 ....(5)....(5)

λ2(5x1+2x210)=0.....(6)\lambda_2(5x_1+2x_2-10)=0\>.....(6)



Non-negativity constraints


λ10,  λ20  .....(7)\lambda_1\ge0,\>\>\lambda_2\ge0\>\>.....(7)



Case (i)


λ1=0,λ2=0\lambda_1=0,\lambda_2=0

From (1) and(2), 22x1=0     x1=12-2x_1=0\implies\>x_1=1

34x2=0     x2=343-4x_2=0\implies\>x_2=\frac{3}{4}


Point (1,34)(1,\frac{3}{4}) is a KKT point


Case (ii)


λ1=0,λ20\lambda_1=0,\lambda_2\ne0

Using (1),(2) and (6)

22x1+5λ2=02-2x_1+5\lambda_2=0

34x2+2λ2=03-4x_2+2\lambda_2=0

5x1+2x210=05x_1+2x_2-10=0


x1=2+5λ22,  x2=3+2λ24x_1=\frac{2+5\lambda_2}{2},\>\>x_2=\frac{3+2\lambda_2}{4}


5(2+5λ22)+2(3+2λ24)=05(\frac{2+5\lambda_2}{2})+2(\frac{3+2\lambda_2}{4})=0


54λ2=1454\lambda_2=14

λ2=727\lambda_2=\frac{7}{27}


x1=2+5(727)2=8954x_1=\frac{2+5(\frac{7}{27})}{2}=\frac{89}{54}


x2=3+2(727)4=95108x_2=\frac{3+2(\frac{7}{27})}{4}=\frac{95}{108}


This could be a KKT point


Case(iii)


λ10,λ2=0\lambda_1\ne0,\lambda_2=0

22x1+λ1=02-2x_1+\lambda_1=0

34x2+3λ1=03-4x_2+3\lambda_1=0

x1+3x26=0x_1+3x_2-6=0


x1=2+λ12x_1=\frac{2+\lambda_1}{2} x2=3+3λ14x_2=\frac{3+3\lambda_1}{4}


2+λ12+3(3+3λ1)4=6\frac{2+\lambda_1}{2}+\frac{3(3+3\lambda_1)}{4}=6

     λ1=1\implies\>\lambda_1=1


Substituting in x1x_1 and x2x_2

x1=32,  x2=32x_1=\frac{3}{2},\>\>x_2=\frac{3}{2}


But 5x1+2x2=10.55x_1+2x_2=10.5

This violate ......(4)


(32,32)\therefore(\frac{3}{2},\frac{3}{2}) is not a KKT point.



Case (iv)


λ10,λ20\lambda_1\ne0,\>\lambda_2\ne0


22x1+λ1+5λ2=0......(i)2-2x_1+\lambda_1+5\lambda_2=0\>......(i)

34x2+3λ1+2λ2=0....(ii)3-4x_2+3\lambda_1+2\lambda_2=0\>....(ii)

x1+3x2=6.....(iii)x_1+3x_2=6\>.....(iii)

5x1+2x2=10.....(iv)5x_1+2x_2=10\>.....(iv)



Solving (iii) and (iv)


5x1+15x2=305x_1+15x_2=30

5x1+2x2=105x_1+2x_2=10

    13x2=20\implies13x_2=20

x2=2013x_2=\frac{20}{13}

x1=1813x_1=\frac{18}{13}


Substituting x1x_1 and x2x_2 in (i) and (ii)


λ1+5λ2=1013\lambda_1+5\lambda_2=\frac{10}{13}

3λ1+2λ2=41133\lambda_1+2\lambda_2=\frac{41}{13}


Solving

3λ1+15λ2=30133\lambda_1+15\lambda_2=\frac{30}{13}

3λ1+2λ2=41133\lambda_1+2\lambda_2=\frac{41}{13}

λ2=11169,λ1=185169\lambda_2=\frac{-11}{169},\>\lambda_1=\frac{185}{169}


λ2<0\lambda_2<0

This violates (7)


z=2x1+3x2x122x22z=2x_1+3x_2-x_1^2-2x_2^2


Substituting point (1,34)(1,\frac{3}{4})


z=2(1)+3(34)(1)22(34)2z=2(1)+3(\frac{3}{4})-(1)^2-2(\frac{3}{4})^2

=2.125=2.125



Substituting point (8954,95108)(\frac{89}{54},\frac{95}{108})


z=2(8954)+3(95108(8954)22(95108)2z=2(\frac{89}{54})+3(\frac{95}{108}-(\frac{89}{54})^2-2(\frac{95}{108})^2


=1.671296=1.671296


Minimum z=1.671296z=1.671296

at point (8954,95108)(\frac{89}{54},\frac{95}{108})








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