sing levar opt
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STEPS IN FORMULISATION OF ANOPTIMISATION PROBLEM
NEED IDENTIFICATION
CHOOSE DESIGN VARIABLES
FORMULATE CONSTRAINTS
FORMULATE OBJECTIVE FUNCTION
SET UP VARIABLE BOUNDS
CHOOSE OPTIMISATION ALGORITHM
OBTAIN SOLUTION(S)
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CLASSICAL OPTIMIZATIONTECHNIQUES
SINGLE-VARIABLE OPTIMIZATION
MULTI-VARIABLE OPTIMIZATION
- WITH NO CONSTRAINTS
- WITH EQUALITY CONSTRAINTS
- WITH INEQUALITY CONSTRAINTS
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FUNCTION IS SAID TO HAVE
at if *xx =- RELATIVE OR LOCALMINIMUM
)*(*)( hxfxf +
- RELATIVE OR LOCALMAXIMUM
at if *xx = )*(*)( hxfxf +For all sufficiently small positive and negative values of h
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FUNCTION IS SAID TO HAVE
at if *x- ABSOLUTE OR GLOBAL MINIMUM
)(*)( xfxf
- ABSOLUTE OR GLOBAL MAXIMUM
at if *x )(*)( xfxf for all x in the domain defined for )(xf
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Single variable optimization Bracketing (exhaustive search)
Region elimination (internal halving)
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Step 3 Is ? If yes, go
to Step 2;Else no minimum
exists in (a,b) or a boundarypoint (a or b) is the minimum
point.
Exhaustive search algorithm (given f(x), a & b)
Step 1 set = a, x = (b-a)/n (n is the number ofintermediate points), and .
Step 2 If , the minimum point lies in ,Terminate;
Else , and go to Step 3.
1x
,12 xxx += xxx += 23
( ) ( ) ( )321 xfxfxf ( )31,xx
xxxxxxx +=== 233221 ,,
bx 3
a 1x 2x 3x ..... b
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Step 1 Choose a lower bound a and an upper bound b. Choosealso a small number. Let ,Compute
Step 2 Set Compute and
Step 3 If set go to Step 5;
Else go to Step 4.
Interval Halving Method
( ) 2baxm += .abLLo ==).(
mxf
,4/1 Lax += .4/2 Lbx = ).( 2xf)( 1xf
)()( 1 mxfxf < 1; xxxb mm ==
Step 4 If f(x2) < f(xm) {a = xm; xm = x2; go to step 5}Else {a = x1; b = x2; go to step 5}.
Step 5 Calculate L = b-a If |L| < Terminate;Else go to Step 2.
a1
x2
xm
x b
L
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EXAMPLE Minimize in the interval (0,5).
Step 1: ; a=0; b=5; ; ;
Step 2:
Step 3: IS NO. Step 4: IS NO.
hence [1.25 - 3.75] i.e a=1.25; b=3.75.
Step 5: L=2.5; a=1.25;b=3.75;
xxxf 54)( 2 +=
310= 50 =L 5.2=mx .85.27)( =mxf
75.3;25.1 21 == xx
4.28)(;7.44)( 21 == xfxf
?)()( 1 mxfxf
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