chapter 16. calculus methods of optimization
TRANSCRIPT
Min Soo KIM
Department of Mechanical and Aerospace EngineeringSeoul National University
Chapter 16. Calculus Methods of Optimization
Optimal Design of Energy Systems (M2794.003400)
Chapter 16. Calculus Methods of Optimization
16.1 Continued exploration of calculus methods
- A major portion of this chapter deals with principles of calculus method
- Substantiation for the Largrange multiplier equations will be provided
- Physical interpretation of λ, and test for maxima-minima are also provided
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Chapter 16. Calculus Methods of Optimization
16.1 Continued exploration of calculus methods
- Lagrange multipliers (from Chap.8)
1 2( , , , )ny y x x x
1 1 2
1 2
( , , , ) 0
( , , , ) 0
n
m n
x x x
x x x
1
0m
i i
i
y
1 1 2
1 2
( , , , ) 0
( , , , ) 0
n
m n
x x x
x x x
n+m scalarequations
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Chapter 16. Calculus Methods of Optimization
16.2 The nature of the gradient vector ( )
1. normal to the surface of constant y
1 1 2 2 3 3ˆ ˆ ˆdx i dx i dx i
1 2 3
1 2 3
0y y y
dy dx dx dxx x x
2 2 3 31
1
( / ) ( / )
/
y x dx y x dxdx
y x
2 2 3 31 2 2 3 3
1
( / ) ( / ) ˆ ˆ ˆ/
y x dx y x dxT i dx i dx i
y x
0T y
1 2 3
1 2 3
ˆ ˆ ˆy y yy i i i
x x x
For arbitrary vector :
To be tangent to the surface :
Tangent vector :
Gradient vector :
→ gradient vector is normal to all tangent vectors
→ gradient vector normal to the surface
y
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Chapter 16. Calculus Methods of Optimization
16.2 The nature of the gradient vector ( )
2. indicate direction of maximum rate of change of y with respect to x
1 2
1 2
n
n
y y ydy dx dx dx
x x x
2 2 2 2
1 2( ) ( ) ( )ndx dx dx r constraints :
to find maximum dy :
y
The constraint indicates a circle of radious for 2 dimensions, and a sphere for 3 dimensions
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Chapter 16. Calculus Methods of Optimization
16.2 The nature of the gradient vector ( )
12 0
2i i
i i
y ydx dx
x x
using Lagrange multipliers :
1 1 2 2 1 2
1 2
1 1ˆ ˆ ˆ ˆ ˆ ˆ2 2
n n n
n
y y ydx i dx i dx i i i i y
x x x
y
In vector form :
→ indicates the direction of maximum change for a given distance in the spacey
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Chapter 16. Calculus Methods of Optimization
16.2 The nature of the gradient vector ( )
3. points in the direction of increasing y
y
small move in the x1-x2 space : 1 2
1 2
y ydy dx dx
x x
2 2
1 2
0y y
dy cx x
If the move is made in the direction of :y
1 1 2 2( / ), ( / )/
i
i
dxc dx c y x dx c y x
y x
→ dy is equal or greater than zero
→ y always increase in the direction of y
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Optimize subject to
Chapter 16. Calculus Methods of Optimization
16.5 Two variables and one constraint (prove Lagrange multiplier method)
1 2
1 2
y yy x x
x x
21 2 1 2
1 2 1
/0
/
xd x x x x
x x x
22
1 1 2
/
/
xy yy x
x x x
1 2( , )y x x 1 2( , ) 0x x
Taylor expansion :
substituting the result for ∆𝑥1
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In order for no improvements of Δy,
Chapter 16. Calculus Methods of Optimization
16.5 Two variables and one constraint (prove Lagrange multiplier method)
2 2
0y
x x
1 1
0y
x x
∆𝑦 = −𝜕𝑦
𝜕𝑥1
Τ𝜕𝜙 𝜕𝑥2Τ𝜕𝜙 𝜕𝑥1
+𝜕𝑦
𝜕𝑥2∆𝑥2 = 0 −
𝜕𝑦
𝜕𝑥1
Τ𝜕𝜙 𝜕𝑥2Τ𝜕𝜙 𝜕𝑥1
+𝜕𝑦
𝜕𝑥2= 0
If λ is defined as 𝜕𝑦𝜕𝑥1𝜕𝜙𝜕𝑥1
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Chapter 16. Calculus Methods of Optimization
16.6 Three variables and one constraint
As a same manner with 2 variables, 𝑦 𝑥1, 𝑥2, 𝑥3 is need to be optimized subject to the constraint, 𝜙 𝑥1, 𝑥2, 𝑥3
The first degree terms in the Taylor seriers are
𝜕𝑦
𝜕𝑥1∆𝑥1 +
𝜕𝑦
𝜕𝑥2∆𝑥2 + (
𝜕𝑦
𝜕𝑥3)∆𝑥3
𝜕𝜙
𝜕𝑥1∆𝑥1 +
𝜕𝜙
𝜕𝑥2∆𝑥2 + (
𝜕𝜙
𝜕𝑥3)∆𝑥3
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Chapter 16. Calculus Methods of Optimization
16.6 Three variables and one constraint
Any two of the three variables can be moved independently, but the motion of the third variables must abide by the constraint. Arbitraily choosing 𝑥2 as a dependant one,
∆𝑦 =𝜕𝑦
𝜕𝑥1∆𝑥1 −
𝜕𝑦
𝜕𝑥2
Τ𝜕𝜙 𝜕𝑥1Τ𝜕𝜙 𝜕𝑥2
∆𝑥1 +Τ𝜕𝜙 𝜕𝑥3Τ𝜕𝜙 𝜕𝑥2
∆𝑥3 +𝜕𝑦
𝜕𝑥3∆𝑥3 = 0
In order for no improvement to be possible
𝜕𝑦
𝜕𝑥1−
𝜕𝑦
𝜕𝑥2
𝜕𝜙𝜕𝑥1𝜕𝜙𝜕𝑥2
= 0𝜕𝑦
𝜕𝑥3−
𝜕𝑦
𝜕𝑥2
𝜕𝜙𝜕𝑥3𝜕𝜙𝜕𝑥2
= 0
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Chapter 16. Calculus Methods of Optimization
16.6 Three variables and one constraint
Define 𝜆 as𝜕𝑦𝜕𝑥1𝜕𝜙𝜕𝑥1
𝜕𝑦
𝜕𝑥1− 𝜆
𝜕𝜙
𝜕𝑥1= 0
Then𝜕𝑦
𝜕𝑥2− 𝜆
𝜕𝜙
𝜕𝑥2= 0
𝜕𝑦
𝜕𝑥3− 𝜆
𝜕𝜙
𝜕𝑥3= 0
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Chapter 16. Calculus Methods of Optimization
16.6 Three variables and one constraint
Define 𝜆 as𝜕𝑦𝜕𝑥1𝜕𝜙𝜕𝑥1
𝜕𝑦
𝜕𝑥1− 𝜆
𝜕𝜙
𝜕𝑥1= 0
Then𝜕𝑦
𝜕𝑥2− 𝜆
𝜕𝜙
𝜕𝑥2= 0
𝜕𝑦
𝜕𝑥3− 𝜆
𝜕𝜙
𝜕𝑥3= 0
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Chapter 16. Calculus Methods of Optimization
16.8 Alternate expression of constrained optimization problem
optimize 1 2( , )y x x
1 2( , )x x b subject to
1 2 1 2 1 2( , ) ( , ) [ ( , ) ]L x x y x x x x b unconstrained function
optimum occurs where 0L
1 1 1
2 2 2
1 2
0
0
( , ) 0
L y
x x x
L y
x x x
x x b
find optimum point
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Chapter 16. Calculus Methods of Optimization
16.9 Interpretation λ of as the sensitivity coefficient
* * *
1 21 2
1 2
( , ) (1)x xy y y
y x x SCb x b x b
1 21 2
1 2
( , ) 1 0 (2)x x
x x bb x b x b
* * * *
1 2
* *
1 2
( / ) ( / )
( / ) ( / )
y x y x
x x
1 2
1 2
(2) 0x xy y
x b x b
sensitivity coefficient (SC) :
(1) SC
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