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Unit 1

PART - A

1. State the necessary and sufficient conditions for the Maximum of a multi variablefunction f(X).

2. Define a saddle point and indicate its significance.

3. How do you test the ositive! "egative and #ndefiniteness of a S$uare

Matrix %&'

4. ind the Maximum of a function f (X) * +x, - x+ -, sub/ect to g(X) * x,- +x++* 0

using 1agrange multiplier method.

5. or a problem of two variables with one constraint! explain the method of findingsolution by 1agrange multiplier method.

6. Define the following2

(i) easible solution

(ii) 3asic solution

(iii) 3asic feasible solution

(iv) "on degenerative 3asic feasible solution

(v) 4ptimal 3asic solution.

7. Define degeneracy and cycling in a Simplex roblem and explain the method toresolve the degeneracy.

8. 5hat is the need for &rtificial variable 6echni$ue 1ist the disadvantage of

3ig 7 M method over 6wo7hase method.

PART - B

1. (i) Determine the maximum and minimum value of the function

f(x) * ,+x89 :8 x: - :x0 - 8

(ii) Minimi;e f(x! y) * +

7+x, - x+ 7 8x0 ? 7 @

:x, - x+ - x0 > @

xi ? i * ,! +! 0

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using simplex method

3. Ase the simplex method to solve the 1

Maximi;e ; * 0x, - 8x+

sub/ect to the constraints

x, - +x+ > +

+x, - x+ > +

x+ > @ and

x,! x+?

4. Ase the simplex method to solve the 1

Minimi;e ; * 7x, - x+ 7 0x0

sub/ect to the constraints

x, - x+ - x0 > ,

7 +x, - x0 ? 7 +

+x, 7 +x+ - 0x0 > and

x,! x+! x0 ?

5. Ase 3ig 7 M to

Minimi;e ; * :x, - x+

sub/ect to the constraints

0x,- x+ * 0

:x, - 0x+ ? @x,- +x+ > 0

x,! x+ ?

6. Ase two phase simplex method to

Minimi;e ; * +x, - 0x+ - +x0 7 x: - x8

sub/ect to the constraints

0x, 7 0x+ - :x0 - +x: 7 x8 *

x, - x+ - x0-0x:- x8 * +

xi ? i * , to 8

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UNIT- II

PART - A

1. (i) 5hat is an one7dimensional minimi;ation problem

(ii) 5hat are the limitations of classical methods in solving a one7 dimensional

minimi;ation problem

2. 1ist out various one 9dimensional minimi;ation methods

3. 5rite the search &lgorithm with a fixed step si;e to find the minimum of afunction.

4. Bxplain Bxhaustive algorithm used to find minimum of a function.

5. (i) 5hat is the difference between ibonacci and golden section methods

(ii) 1ist out the limitations of ibonacci method

6. Bxplain the ibonacci method to find the minimum of an one variable function

7. Bxplain the golden section method to find the minimum of a function

8. Compare various Blimination methods vi;! Bxhaustive search! Dichotomoussearch! ibonacci and golden section methods.

PRAT - B

1. Minimi;e f(x) * .@8 9 %.8 = (,- x+)' 9 .@8 (x tan7,(,=x)) in the interval %!0' bythe ibonacci method using n * @

2. ind the minimum of the function f * 8 7 8E0 7 +E - 8 by Bxhaustive search inthe interval (!8).

3. Minimi;e f(x) * x (x 7 ,.8) in the interval (!,) by Folden section method using

n * .

4. ind the maximum of the function

f(E) * ,= % +(,-E+),=+' 7 (,-E+),=+(,7 (,= % +(,-E+) ' ) ) - E using "ewton methodwith staring point .@

5. Minimi;e f(x) * x - :=x using ibonacci search in the interval (!+) using n * @

6. ind the minimum of f * x - :x 7,in the interval (! +) to within , G of the exactvalue.

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UNTI - III

PART - A

1. (i) State the necessary and sufficient conditions for the unconstrained minimum

of a function

(ii) Five three reasons why the study of unconstrained minimi;ation is important.

2. Bxplain with a flowchart! the iterative procedure used in unconstrainedoptimi;ation.

3. Bxplain univariate method in finding the minimum of a function.

4. Bxplain the andom wal< method in detail

5. Bxplain Hoo