Optimal Design of Energy Systems

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Optimal Design of Energy Systems

Chapter 8 Lagrange Multipliers

Min Soo KIM

Department of Mechanical and Aerospace Engineering

Seoul National University

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8.1 Calculus methods of optimization Function – differentiable

Constraints - equality

Chapter 8. Lagrange Multipliers

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8.2 Lagrange Multiplier equations optimize

subject to

→ optimum occurs at x values

Chapter 8. Lagrange Multipliers

1 2

( , , , _n ) y  y x x  x

1 1 1 2

1 2

( , , , ) 0

n

m m n

x x x

 

 







1 1 _m _m 0

y

   

       

constants : Lagrange multipliers

1 2

ˆ ˆ ˆ

n n

y y y

y i i i

x x x

  

    

   

unit vectors Gradient vector : normal to y=const. surface

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8.2 Lagrange Multiplier equations

* *

1 _m , x1 x_n



_



_

at optimum point

1

1 1

1 1 1

ˆ : y _m ^m 0

i x x x





^

 

^

    

   

1

ˆ :_n 1 _m ^m 0

n n n

i y

x x x

 



^



^

    

   

• n scalar equations

• m constraints equations

• unknowns

m

n

m  n m  n

if fixed values of x’s (no optimization)



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8.5 Unconstrained optimization

1 2

( , , , _n ) y  y x x  x

0

 y

→

1 2

0

n

y y y

x x x

      

   

ex) minimize (optimize)

^y ^ ^x²

2 0

y x

x

  



2 2

1 2

y  x  x

1 1

2 2

2 0

y x

x

y x

x

  



  



x1

x2

y

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8.6 Constrained optimization

x1

x2

y

ex)

minimize

constraint .:

→

 Lagrange multiplier

2 2

1 2

y  x  x

1 2 1

x  x 

1 2 1 1 2 2

1 2

ˆ ˆ 2 ˆ 2 ˆ

y y

y i i x i x i

x x

 

    

 

1 2 1 0

x x



   

1 2 1 2

1 2

ˆ ˆ ˆ ˆ

i i i i

x x

 



^ ^

    

 

1 ˆ1 2 ˆ2

( 2 ) ( 2 ) 0

y

 

x



i x



i

       

1 2

2 0

x x



 

1 2

1 2 m in

/ 2, / 2

0 .5, 0 .5 0 .5

x x

x x y

 

 

   

→

x1

x2

+

x1

x2

y

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8.6 Constrained optimization

<Example 8.3> A total length of 100 m of tubes must be installed in a shell- and-tube heat exchanger, in order to provide the necessary heat- transfer area.

Total cost of the installation in dollars includes

1. The cost of the tubes, which is constant at $ 900 2. The cost of the shell =1100D^2.5L

3. The cost of the floor space occupied by the heat exchanger=320 DL

Chapter 8. Lagrange Multipliers

where L is the length of the heat exchanger and D is the diameter of the shell, both in meters.

The spacing of the tubes in such that 200 tubes will fit in a cross-sectional area of 1 m² in the shell. Determine the diameter and length of the heat exchanger for minimum first cost.

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8.6 Constrained optimization

minimize subject to

9 0 0 1 1 0 0 2 .5 3 2 0

y   D L  D L

2

2 2

2

( ) ( ) 2 0 0 ( / ) 1 0 0 ( )

4

5 0 1 0 0

D m L m tu b es m m

D L



  

 

y

 

   ˆ1

: D i

ˆ2

: L i

( 2 7 5 0D1 .5L  3 2 0 )L 



(1 0 0



D L)  0

2 .5 2

(1 1 0 0D  3 2 0D) 



(5 0



D )  0

2 7 5 0 1 .5 3 2 0 1 0 0

D

 D



 

* *

0.7 , 1.3 , 8.78

D  m L  m





* 2.5

900 (1100)(0.7) (1.3) (320)(0.7)(1.3) $1777.45

y    

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8.7 Gradient vector

- gradient vector is normal to contour line

x2

x1

y  y3

y  y2

y  y1

d x1

d x2

1 2

( , )

y  y x x

1 2

y y

d y d x d x

x x

 

 

 

0

d y   ₁ ₂ ²

1

/ /

y x

d x d x

y x

 

   

substitution into arbitrary unit vector

1 2

2 2

1 2

ˆ ˆ

( ) ( )

d x i d x j

d x d x



Along the line of ^y ^ ^const^.

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8.7 Gradient vector

Tangent vector

Gradient vector

2

1 2 1 2 1

2 2 2 2 2

1 2 2

1 1 2

/ ˆ ˆ ˆ ˆ

ˆ ˆ /

( ) ( ) /

/ 1

y x y y

i j i j

d x i d x j y x x x

T

d x d x y x y y

y x x x

   

   

    

  

            



2 1

2 2

1 2

ˆ ˆ

y y

i j

x x

G

y y

x x

  

 



      

     

   

 T  G  0

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8.9 Test for maximum or minimum

- decide whether the point is a maximum, minimum, saddle point, ridge, or valley

saddle point ridge & valley point

ridge

valley

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8.9 Test for maximum or minimum

• expected minimum occurs at

( ) y  y x

2

2 2

( ) ( ) ( ) 1 ( )

2

d y d y

y x y a x a x a

d x d x

      Taylor series expansion near x  a

when

d y

 d x should be zero

2

2 0

d y d x 

If ^{y x}⁽ ¹⁾ ^ ^{y a}^{( )} when ^x¹ ^ ^a ( 2) ( )

y x  y a x₂  a ^minimum

2

2 0

d y d x 

If y x( ¹)  y a( ) x₁  a maximum

( 2) ( )

y x  y a x₂  a

a

d y 0 d x 

If ^{y x}^{( )} ^ ^{y a}^{( )} ^x ^ ^a

( ) ( )

y x  y a when x  a

is still considered minimum ( )

y a

is not minimum ( )

y a

when when when

x1

x2

a x1

x2

x  a

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8.9 Test for maximum or minimum

• expected minimum or maximum occurs at

1 2

( , )

y  y x x

1 2 1 2 1 1 2 2

1 2

2 2

1 1 1 1 1 2 1 1 2 2 2 2 2 2

( , ) ( , ) ( ) ( )

1 1

( ) ( )( ) ( )

2 2

y y

y x x y a a x a x a

x x

y x a y x a x a y x a

 

    

 

  

       

min

1 1 1 2

2 1 2 2

2

1 1 2 2 1 2

y y

D y y

y y y

 

  

  

 

0

D  y1 1  0 (y2 2  0 )

0

D  max

1 2

(a ,a )

and

0

D  not min or max

1 1 0 ( 2 2 0 )

y  y 

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8.9 Test for maximum or minimum

<Example 8.4> optimal value x

₁

and x

₂

& maximum or minimum ?

2 1

2 1 2

2 4

4

y x x

  x x 

1

2

1 1 2

2 2

x y

x x x

  



₂ ₁ ₂²

2 4

y

x x x

   

and



₁^* ₂^*

1

2, 2

x  x 

→

2 2 2

2 2

1 2 1 2

3 1 6 2

2

y y y

x x x x

     

   

1 1

3 / 2 2

0 0

2 1 6

D   y   

_minimum

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8.10 Sensitivity coefficient

- represents the effect on the optimal value of slightly relaxing the constraints

- variation of optimized objective function

(

*

)

8 .7 8 C o st

S C   H   



y

*

In example 8.3, if the total length of the tube increases from 100m to random value ‘H’, what would be the increase in minimum cost ?

* 2 .5

9 0 0 1 1 0 0 3 2 0 9 0 0 8 .7 8

C o st   D L  D L   H

2 2

5 0



D L  1 0 0  5 0



D L  H

Extra meter of tube for the heat exchanger would cost additional $8.78