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Optimisation, Graphical Solution, Bisection method, convergence newton's method, Secant method, method of false position, golden section method
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Want to minimise some function F(
x
) subject
to constraints,
a
( x i^
) = 0, i=1,2,…,m
1
b
( x i^
0, i=1,2,…,m
2
where
x
is a vector of length n.
-^
An oil refinery can buy light crude at £35/barrel andheavy crude at £30/barrel.
-^
Refining one barrel of oil produces petrol, heating oil,and jet fuel as follows:
Petrol
Heating oil
Jet fuel
Light crude
Heavy crude
-^
The refinery has contracts for 0.9M barrels of petrol,0.8M barrels of heating oil and 0.5M barrels of jet fuel.
-^
How much light and heavy crude should the refinerybuy to satisfy the contracts at least cost?
Let x
1
and x
2
be the number of barrels (in
millions) of light and heavy crude that therefinery purchases.
-^
Cost (in millions of £): F(
x
) = 35x
1
2
Constraints:
0.3x
1
2
0.9 (petrol)
0.2x
1
2
0.8 (heating oil)
0.3x
1
2
0.5 (jet fuel)
x
1
0, x
2
0 (non-negativity)
This is calleda “linearprogram”
(x
,x 1
( x
1
2
Recall that:
F(
x ) = 35x
1
2
-^
i.^
f^
£(x*) = 0 (first order condition)
ii.
f^
≥(x*)
¥
0 (second order condition)
f(x) = (x-1)(x-1)+
-^
Evaluate f at mid-point c=(a+b)/2.
If f(c) is zero then quit.
If f(a) and f(c) have the same sign then set a=c;else set b=c.
Go to Step 1.
a^
b
(a+b)/
f=@(x)(2x-4sin(x)); >> a=1; fa=f(a); >> b=2; fb=f(b); >> c=(a+b)/2;fc=f(c);if fa*fc>0 a=c; else b=c; end;c •^
Using the up arrow to repeat the last line we getvalues of c that converge to the solution off(x)=0.
f(x
)k
k^
xk
xk+
f^
£(x
) = f(xk
)/(xk
-xk^
k+
x
k+
= x
k^
)/fk
£ (x
)k^
Convergence of Newton’s Method
k+
2
e
1
If 0<x
<e then 1
Newton’s methodwill converge. If x
e it will fail.
-^
Newton method requires first derivative at each iteration.
-^
The bisection method doesn’t use the magnitudes of f ateach end of the interval.
-^
Suppose we use f(a
) and f(bn
) and finds a new estimaten
of the zero by approximating the function between a
n^
and
b
n^
by a straight line.
f(b
)n
b
n
a
n
xn
f(a
)n
⎞ ⎟⎟ ⎠
⎛ ⎜⎜ ⎝
− −
−
=^
) ( ) ( ) ( n
n
n
n
n
n
n^
a f
b f
a b a f a x
To avoid possible divergence problem withthe secant method we keep the zerobracketed in an interval (a,b), as in thebisection method.
If f(c) = 0 we are finished.
-^
If f(a) and f(c) have the same sign we replacea by c; otherwise, we replace b by c.
