1) gi(x∗)−bi is feasible for i=1,2....m
Where gi(x∗) is optional value
Writing the equation in residue form
f)x)=2x1+3x2−x12−2x22
g1(x)=x1+3x2−6
g2(x)=5x1+2x2−10
Applying the Kuhn-Tucker conditions (Optimality Condition)
2)
⎝⎛∂x1∂f∂x2∂f⎠⎞ +λ1 ⎝⎛∂x1∂g1∂x2∂g1⎠⎞ +λ2 ⎝⎛∂x1∂g2∂x2∂g2⎠⎞=0
2−2x1+λ1+5λ2=0......(1)
3−4x2+3λ1+2λ2=0.....(2)
Feasibility condition
x1+3x2≤6 ......(3)
5x1+2x2≤10......(4)
Complimentary slackness property
λ1(x1+3x2−6)=0 ....(5)
λ2(5x1+2x2−10)=0.....(6)
Non-negativity constraints
λ1≥0,λ2≥0.....(7)
Case (i)
λ1=0,λ2=0
From (1) and(2), 2−2x1=0⟹x1=1
3−4x2=0⟹x2=43
Point (1,43) is a KKT point
Case (ii)
λ1=0,λ2=0
Using (1),(2) and (6)
2−2x1+5λ2=0
3−4x2+2λ2=0
5x1+2x2−10=0
x1=22+5λ2,x2=43+2λ2
5(22+5λ2)+2(43+2λ2)=0
54λ2=14
λ2=277
x1=22+5(277)=5489
x2=43+2(277)=10895
This could be a KKT point
Case(iii)
λ1=0,λ2=0
2−2x1+λ1=0
3−4x2+3λ1=0
x1+3x2−6=0
x1=22+λ1 x2=43+3λ1
22+λ1+43(3+3λ1)=6
⟹λ1=1
Substituting in x1 and x2
x1=23,x2=23
But 5x1+2x2=10.5
This violate ......(4)
∴(23,23) is not a KKT point.
Case (iv)
λ1=0,λ2=0
2−2x1+λ1+5λ2=0......(i)
3−4x2+3λ1+2λ2=0....(ii)
x1+3x2=6.....(iii)
5x1+2x2=10.....(iv)
Solving (iii) and (iv)
5x1+15x2=30
5x1+2x2=10
⟹13x2=20
x2=1320
x1=1318
Substituting x1 and x2 in (i) and (ii)
λ1+5λ2=1310
3λ1+2λ2=1341
Solving
3λ1+15λ2=1330
3λ1+2λ2=1341
λ2=169−11,λ1=169185
λ2<0
This violates (7)
z=2x1+3x2−x12−2x22
Substituting point (1,43)
z=2(1)+3(43)−(1)2−2(43)2
=2.125
Substituting point (5489,10895)
z=2(5489)+3(10895−(5489)2−2(10895)2
=1.671296
Minimum z=1.671296
at point (5489,10895)
Comments
Leave a comment