Chapter 5 One Dimensional Search MEF

Chapter 5

Chapter 5

One Dimensional Search

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Chapter 5

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Unidimensional Search

Chapter 5

(1) If have a search direction, want to minimize in that
direction by numerical methods

(2) Search Methods in General
2.1. Non Sequential – Simultaneous evaluation of f at n

points – no good (unless on parallel computer).
2.2. Sequential – One evaluation follows the other.
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Chapter 5

(3) Types of search that are better or best is often
problem dependent. Some of the types are:
a. Newton, Quasi-Newton, and Secant methods.
b. Region Elimination Methods (Fibonacci, Golden
Section, etc.).
c. Polynomial Approximation (Quadratic Interpolation,
etc.).
d. Random Search
(4) Most methods assume
(a) a unimodal function, (b) that the min is
bracketed at the start and (c) also you start in a
direction that reduces f.

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Chapter 5

To Bracket the Minimum

1. Compute f ( x( 0) ) and f ( x0  x)

If f ( x( 0)  x)  f ( x( 0) ), let xNEW  xOLD
If f ( x( 0)  x)  f ( x( 0) ), let xNEW  2xOLD

2. Compute f ( x(1) )  f ( x( 0)  xNEW )

Continue doubling x until f ( x( 0)  x( k ) )  f ( x( 0)  x( k 51) )

3. Compute f ( x( k 1)  x( k 2) )

Chapter 5

1
(2 )

2

You now have 4 equally spaced points x(3) , x , x( 2) , x(1) .
Discard the point that enables you to keep a bracket
on the point giving the minimum f(x) (using the
closest points)

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Chapter 5

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1. Newton’s Method
Newton’s method for an equation is

Chapter 5

f ( x)  f ( x0 )  f ( x0 )( x  x0 )  0

f ( x0 )
( x  x0 )  
f ( x0 )

or

f ( x0 )
x  x0 
f ( x0 )

Application to Minimization
The necessary condition for f(x) to have a local minimum
is f (x) = 0. Apply Newton’s method.

x( k 1)

(k)

f (x )
(k)

x 
(k)


f (x )

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Examples
Minimize

f ( x)  a 0  a1 x  a 2 x2

Chapter 5

f ( x)  a1  2a 2 x
f ( x)  2a 2

x

(1)

x

a 1  2 a 2 x( 0 )
a1
a1
(0)
(0)

 x (
x )
2a 2
2a 2
2a 2

(0)

Minimize

f ( x)  x4  x2  1
f ( x)  4 x3  2 x

f ( x)  12 x2  2
x

(1)

x

(0)

Start at x

3 x4  2 x

12 x2  2

(0)

Continue

3 2
 1, x  1 
 0.100
12  2
(1)

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Advantages of Newton’s Method

Chapter 5

(1) Locally quadratically convergent (as long as f (x) is
positive – for a minimum).
(2) For a quadratic function, get min in one step.

Disadvantages
(1) Need to calculate both f (x) and f (x)

(2) If f (x)→0, method converges slowly
(3) If function has multiple extrema, may not converge
to global optimum.

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2. Finite-Difference Newton Method
Replace derivatives with finite differences

Chapter 5

x( k 1)

f ( x  h)  f ( x)
h
 x( k ) 
f ( x  h)  2 f ( x)  f ( x  h)
h2

Disadvantage

Now need additional function evals (3 here vs. 2 for Newton)

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3. Secant(Quasi-Newton) Method

f ( x( k ) )  m( x  x( k ) )  0

Chapter 5

Analogous equation to (A) is

( B)

The secant approximates f (x) as a straight line

f ( x ( q ) )  f ( x ( p ) )
m
x(q )  x( p )
(k)


f
(
x
)
( k 1)
(k)
x 
x
f ( x ( q ) )  f ( x ( p ) )
12
x(q )  x( p )

Start the Secant method by using 2 points spanning x at
which first derivatives are of opposite sign.

Chapter 5

For next stage, retain either x(q) or x(p) so that the pair of
derivatives still have opposite sign.

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Order of Convergence
Can be expressed in various ways. Want to consider how

Chapter 5

x

(k)

 x as k  
*

Linea r
x

( k 1)

x

(k)

x

*

x

*

c

0  c 1

usually slow in practice

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Order P
x

( k 1)

Chapter 5

x

(k)

x

*

x

* p

c

c0

p 1

Fastest in practice
If p = 2, quadratic convergence
p = 1.32 ?

Superlinear
x

( k 1)

lim

k 

x

(k)

x

*

x

*

0

(r  ck a nd ck  0 a s k  )

Usually fast in practice
Some methods can show theoretically what
the order is.

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Chapter 5

Quadratic Interpolation

Approximate f(x) by a quadratic function.
Use 3 points

f ( x)  a  bx  cx 2

Minimize : f ( x)  0  b  2cx so x*  

b
2c
Evaluate f(x) at the 3 points f(x1 ), f ( x2 ), f ( x3 )

Solve 3 simultaneo us equations for a, b, c
f ( x1 )  a  bx1  cx12

f ( x2 )  a  bx2  cx22

f ( x3 )  a  bx3  cx

2
3

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Chapter 5

b

1
1
1

f ( x1 )

x12

f ( x2 )

x

f ( x3 )

2
2
2
3

x

1 x1
1 x2
1 x3

2
1
2
2
2
3

x

x
x

c

1 x1
1 x2
1 x3
1 x1
1 x2
1 x3

f ( x1 )
f ( x2 )
f ( x3 )
x12
x22
x32

(or use Gaussian elimination)

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b  1  f ( x2 ) x32  f ( x3 ) x22   1  f ( x1 ) x32  f ( x3 ) x12 

Chapter 5

1  f ( x1 ) x22  f ( x2 ) x12  :



Numerator



c  f ( x1 )   x32  x22   f ( x2 )  x32  x12   f ( x3 ) (  x22  x12 ) 

 f ( x1 )( x22  x32 )  f ( x2 )( x32  x12 )  f ( x3 )( x  x22 ) :

Denominator
b
x*  
2c
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