second semester model questions
MA602
NON LINEAR PROGRAMMING PROBLEM
UNIT I
1.define (i) unimodel function (ii)positive
semi_definite matrix and negative definite matrix
2. write a short notes on the followings:
(i) convex function (ii) concave function
3.starting from(0,0) use newtons method to
minimize
f(x)=3x12-4x1x2+12x22+4x1+6
4.define (i)unimodel function(ii)convex
function(iii)hessian matrix
Test whether the function f(x)=2x12+2x22+4x32+2x1x3+4x2x3 is concave or
convex
5. use cubic interpolation method to find the
minimum of f(x)=x4-14x3+60x2-70x
6. (a)with necessary derivation explain
fibonacc i method and its limitations
(b) starting from (0,0,0)use newtons method to
minimize f(x)=4x12+3x22+5x32+6x1x2+x1x3-3x1-2x2+1
7. (a)define (i) unimodel function (ii) hessian
matrix(iii)positive semi_definite matrix and negative definite matrix
Test whether the function
f(x)=2x12+2x22+4x32+2x1x3+4x2x3
is concave or convex
(b)using fibonacci method solve the problem
Maximize
F(x)={x/2 for x<=2
-x+3 for x>2
In the interval(0,3). Take N=6
(OR)
(a)Using quadratice interpolation method
,minize
F(x)=x5-5x3-20x+5 in the
interval(0,5)
(b)starting from(0,0)use newtons method to minimize
the function
F(x)=x1-x2+2x12+2x1x2+x22
8.(a)define (i) unimodel function (ii) hessian
matrix(iii)positive semi_definite matrix and negative definite matrix
(iv)convex function
(b)using cubic interpolation method minimize
F(x)=x5-5x3-20x+5 in the interval
(0,5)
(OR)
(a)Find the minimum of the function f(x)=2x2-ex by fibonaccis
method in the range(0,1) take N=10
(b) with necessary derivations outline the
method of quadratic interpolation
9.(a)find sthe maximum of the function
f(x)=x(1.5-x) by fibonacci method in the interval (0,1) with N=10
(b) describe briefly trhe method of cubic
interolation fo unconstrained non_linear rogramming problem
10. (a)define (i)unimodel function(ii)convex
function(iii)hessian matrix.test whether the function f(x)=-x12-6x22-23x32-4x1x2+6x1x3+20x2x3
is convex or concave
(b)using fibonacci method solve the problem
Maximize
F(x)={x/2 for x<=2
-x+3 for x>2
In the interval(0,3). Take N=6
(or)
(a) starting from (0,0,0)use newtons method to
minimize f(x)=4x12+3x22+5x32+6x1x2+x1x3-3x1-2x2+1
b)using cubic interpolation method minimize
F(x)=x5-5x3-20x+5 in the
interval (0,5)
Unit II
1.minimize f(x)=3x12-4x1x2+12x22+4x1+6 by the univariate method.The starting point may be (0,0) and ε=0.01.
2.Using Davidon – Fletcher – powell method
minimize f(x) = 3(x-1)2+2(x2-2)2+(x3-3)2 taking X1=[9,-7,11]T.
3.Outline the method of Nelder and meads and
write down its algorithm.
4.By Davidon – Fletcher – Powell
method,minimize f(x)=x12+x22+x32-4x1-8x2-12x3+100 with the starting point x1={0.0.0}T..
5.(a).Minimize f(x)=3x12-4x1x2+12x22+4x1+6 by the univariate method.The starting point may be (0,0) and ε=0.01
6.By Davidon – Fletcher – powell method
,Minimize f(x)=x1-x2+2x12+2x1x2+x22 With the starting point x1={0 0}.
7.(a).Describe the method of hooke and Jeeves
and write down its algorithm.
(b).Minimize f(x)=3(x1-4)2+5(x2+3)2+7(2x3+1)2 Using Fletcher –
Reeves method with the starting point(0,0,0).
8.(a).Minimize f(x)=3x12+12x22-4x1x2+4x1+6 by the univariate method.The starting point may be (0,0)and ε=0.01.
(b).Minimize f(x)=3(x1-4)2+5(x2+3)2+7(2x3+1)2 using the method
of Fletcher – Reeves,The starting point is(0,0,0).
9.(a).Outline the method of Nelder and Meads
and write its algorithm .
(b). Minimize f(x)=3x12-4x1x2+12x22+4x1+6 using the method of Davidon – Fletcher – Powell ,stqrting
from(0,0).
10.(a)Minimize the function f(x)=(x12+x2-11)2+(x1+x22-7)2 from the starting
point (0,0)using univariate method.Take ε=0.01.
(b).Using Fletcher –Reaves method ,minimize
f(x)=x13+x22-3x1-2x2+2. The starting point is (0,0).
11.(a)Minimize f(x)= 4x12+3x22-5x1x2-8x1
by the univariate method .The starting point may be (0,0)and ε=0.01.
(b).Minimize f(x)=4x12+3x22+5x32+6x1x2+x1x3-3x1-2x2+15 using the method of Fletcher –Reeves .The starting point
is(0,0,0).
12.(a) Outline the method of Nelder and Meads
and write its algorithm .
(b). a)Minimize the function f(x)= x1-x2+2x12+2x1x2+x22 by the
univariate method .The starting point may be (0,0)and ε=0.01.
13.(a).write down the algorithm of Powells
method for multidimensional unconstrained optimization problems.
(b).Minimize f(x)]=4(x1-5)2+(x2-6)2 using Fletcher
–Reeves method.The starting point is(0,0).
14.write down the algorithm of steepest descent
method. Minimize f(x)=(x1-2)2+(x2-5)2+(x3+2)4
by using the steepest descemt ,etjpd with the starting point(4,-2,3)
15. write down the algorithm of hooke and
jeeves method for multidimensional unconstrained optimization problems
16.describe the method of hooe and jeeves and
write down its algorithm
17.prove that the gradient vec tor represents
the direction of steepest ascent
UNIT III
1.write down the algorithm of interior penalty
function method
2.use kuhn_tucker conditions to solve the
following non_linear programming problem
Max z=2x1-x12+x2
Subject to the constraints
2x1+3x2<=6
2x1+x2<=4
X1,x2>=0
3.using the khuntucker conditions optimixe
f(x)=2x1+3x2-x12-x22-x32
subject to
x1+x2<=1
2x1+3x2<=6
X1,x2>=0
4.describe the method of exterior penalty
function for constrained non linear programming problem and write down its
algorithm
5.using kuhntucker conditions optimize
f(x)=7x12+6x1+5x2
Subject to
X1+2x2<=10
X1-3x2<=9
X1,x2>=0
6. by applying interior penalty function method
Minimize f(x)={(x1+1)3/3}+x2
Subject to
G1(x)=1-x1<=0
G2(x)= -x2<=0
X1,x2>=0
7.using the method of lagrangian multipliers,
find the optimum solution of
f(x)=4x1+9x2-x12-x22
subject to
4x1+3x2=15
3x1+5x2=14
X1,x2>=0
8.using kuhntucker conditions
Optimize f(x)=2x1+3x2-(x12+x22+x32)
Subject to
X1+x2<=1
2x1+3x2<=6
X1,x2,x3>=0
9. by applying exterior penalty function method
Minimize f(x)=(x1+2)3+(x2+1)3
Subject to
2- x1<=0
1 – x2<=0
X1,x2>=0
10.using the lagrangian multipliers method find
the optimum solution of f(x)=x12+x22+x32 subject to
X1+x2+3x3=2
5x1+2x2+x3=5
X1,x2,x3>=0
11. Derive the necessary and sufficient
condition for the following non linear programming problems
Max z=f(x)
Subject to
G(x)<=b
x>=0
x=(x1,x2,…..xn)
12.usingf the kuhn tucker conditions optimize
F(x)=10x1+4x2-2x12-x22
Subject to
2x1+x2<=5
X1,x2>=0
13. obtain the kuhn _tucker conditions for the
nllp of minimizing z=f(x1,……..xn)>=bi, i=1,2….,m and xj>=0,j=1,2….,n(n>m)
14. using the kuhn tucker conditions optimize
f(x)=7x12+6x1+5x22
Subject to
X1+2x2<=10
X1-3x2<=9
Xi>=0,x2>=0
15. by applying interior and exterior penalty
function methods
Minimize f(x)=1/3(x1+1)3+x2
Subject to
-x1 +1<=0
-x2<=0
UNIT IV
1. Describe briefly the beale`s method for
solving quadratic programming problem
2. using geometric programming solve
Min z=40x1-1x2-1x3-1+40x2x3+20x1x2+10x1x3
X1,x2,x3>=0
3. Apply wolfe`s method for solving the
quadratic programming problem
Max f(x)=4x1+6x2-2x12-2x1x2-2x22
Subject to
X1+2x2<=2
X1,x2>=0
4. beal`s method for solving the quadratic
programming problem
5. explain the method of obtaining solution of
an unconstrained geometric programming problem from differential calculus point
of view
6. It has been decided to shift grain from a ware
house to a factory in an open rectangular box of length x1metre,width x2metre,height x3metre. The bottom,sides and the ends of the box cost
respectivelyRs.80,10 and 20 per square metre. It costs Re.1 for each round trip
of the box. Assuming that the box will have no salvage value find the minimum
cost involved for transporting 80 cubic metres of grain
7. apply wolfe`s method to solve the quadratic
programming problem QP –Dump Prepared by Kalaivani. - 216109008 10
Maximize f(x)=2x1
+x2 -x12
Subject to
2x1 +3x2<=6
2x1+x2<=4
X1>=0
X2>=0
8. Solve the following quadratic programming
problem by beales method
Max z=10x1+25x2- 10x12- x22- 4x1x2
Subject to
X1+2x2+x3=10
X1+x2+x4=9
And x1,x2,x3>=0
(OR)
Solve the problem
Minimize z=80x1x2+40x2x3+20x1x3
8x1-1x2-1x3-1<=1
And x1,x2,x3>=0 by geometric programming problem
9. When n>k+1, solve the problem
Minimizef(x)=5x1x2-1+2x1-1x2+5x1+x2-1
By geometric programming QP –Dump Prepared by Kalaivani. - 216109008 11
UNIT V
1. (a)Write a short note on piecewise linear
approximation of a non_linear function.
(b) A manufacturing company produces two
products:T.V.sets and Refrigferator sets. Sales_prices relationships for
these two products are given below: PRODUCT
|
QUANTITY DEMANDED
|
UNIT PRICE
|
T.V.SETS
|
1500-5P1
|
P1
|
REFRIGERATORSETS
|
3800-10P2
|
P2
|
ALL THE BEST PRACTICE WELL
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