UNCONSTRAINED
OPTIMISATION
MODULE CONVENOR: GANCARLO IANULARDO
UNCONSTRAINED OPTIMISATION
• Maximum and Minimum values
• First and second order conditions
CONDITIONS FOR A MAXIMUM OF A FUNCTION
z  f( x, y )
Conditions for a maximum of a function
•
z  f( x, y )
Look at fig 1. The surface
has a maximum at P.
Consider the iso−y contour APB. In panel (b):
z > 0 to left of P, z = 0 at P, and z < 0 to right of P.
x
x
x
Therefore in panel (c) the graph of z is downward sloping,
x
2
2
so  z is negative. So at P we have z = 0 and  z< 0.
x
x 2
x 2
•
Look at fig 2. By exactly the same reasoning as above, we
conclude that at P we have z = 0 and  2z < 0.
y
y 2
FIGURE 15.1(A): EXAMINING AN ISO-Y SECTION TO
FIND NECESSARY CONDITIONS FOR A MAXIMUM OF Z
= F (X, Y)
Surface has a
maximum at P
FIGURE 1(B & C): FINDING NECESSARY CONDITIONS
FOR A MAXIMUM OF Z = F (X, Y)
FIGURE 2(A): EXAMINING AN ISO-X SECTION TO FIND
NECESSARY CONDITIONS FOR A MAXIMUM OF Z = F
(X, Y)
Surface has a
maximum at P
FIGURE 2 (B & C): FINDING NECESSARY CONDITIONS
FOR A MAXIMUM OF Z = F (X, Y)
Conditions for a minimum of
a function
Conditions for a minimum of a function
z  f( x, y )
• Look at fig 3, where there is a minimum at R.
When we look at the iso−y contour ERF, we see that:
z
z  0 to left of R, z
at R, and x  0 to right of R.

0
x
x
Therefore in panel (c) the graph of z is upward sloping,
x
2
2

z

z > 0.

z
So
is
positive.
So
at
R
we
have
=
0
and
x 2
x 2
x
• Look at fig 4. By exactly the same reasoning as above,
2
z

z > 0.
we conclude that at R we have y = 0 and
y 2
FIGURE 3(A): EXAMINING AN ISO-Y SECTION TO FIND
NECESSARY CONDITIONS FOR A MINIMUM OF Z = F
Surface has a
(X, Y)
minimum at R
FIGURE 3(B & C): FINDING NECESSARY CONDITIONS
FOR A MINIMUM OF Z = F (X, Y)
FIGURE 4(A): EXAMINING AN ISO-X SECTION TO FIND
NECESSARY CONDITIONS FOR A MINIMUM OF Z = F
(X, Y)
Surface has a
minimum at R
FIGURE 4(B & C): FINDING NECESSARY CONDITIONS
FOR A MINIMUM OF Z = F (X, Y)
Collecting results: (rules 1 and 2)
P in fig 1:
MAX:
z  z  0
;
x y
R in fig 3:
MIN:
z  z  0 ;
x y
 2z  0 ;  2z  0
x 2
y 2
 2z  0 ;  2z  0
x 2
y 2
Rules 1 and 2 are necessary but not sufficient for a max/min.
This is because they refer only to variation in x or y separately.
Saddle points
Look at fig 6. From S, variation in only x or only y reduces z, but some
variations in x and y together increase z.
A point such as S is called a saddle point.
FIGURE 6: HOW SEPARATE AND SIMULTANEOUS VARIATIONS IN
X AND Y CAN HAVE DIFFERENT EFFECTS ON Z
From S, any
variation in x or
y separately
reduces z, but
if x and y
increase or
decrease
together, z may
increase.
SADDLE POINT
To preclude a saddle point, we need in addition to rules 1 and 2:
 2z  2z   2z  2z
x 2 y 2 xy y x
(rule 3)
More on saddle points
A point where
z  z  0
x y
2z  2z
2z  2z


and
(rule 3)

x 2 y 2 xy y x
definitely IS NOT a saddle point.
A point where
z  z  0
x y
and
 2 z  2z   2 z  2 z
x 2 y 2 xy y x
reverse of rule 3) definitely IS a saddle point. (rule 4)
(Note: case where
 2z  2z   2z  2z
x 2 y 2 xy y x
remains open)
(the
A SPECIAL CASE
Fig 7 shows one particular type of SP worth noting.
Point S is a MIN for variation in x alone, so therefore
z  0 and 2z  0
x
x 2
Point S is a MAX for variation in y alone, so therefore
z  0 and
y
 2z  0
y 2
Point S is also a saddle point,
2z 2z  2z 2z (rule 4)
x 2 y 2 xy y x
Key feature is that direct 2nd derivatives have opposite signs
Renshaw: Maths for Economics, 4th edition
FIGURE 7: A SADDLE POINT AT S THAT IS A MAXIMUM FOR VARIATION
IN Y ALONE AND A MINIMUM FOR VARIATION IN X ALONE