Post on 23-Jan-2022
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle pointsLevent Kandiller
Industrial Engineering Department
Cankaya University, Turkey
Minima, Maxima, Saddle points – p.1/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Let us remember the properties for maxima, minima andsaddle points when we have scalar functions with twovariables with the help of the following examples.
Minima, Maxima, Saddle points – p.2/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Let f(x, y) = x2 + y2. Find the extreme points:
Minima, Maxima, Saddle points – p.2/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Let f(x, y) = x2 + y2. Find the extreme points:
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
10
0.5
1
1.5
2
Minima, Maxima, Saddle points – p.2/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Let f(x, y) = x2 + y2. Find the extreme points:
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
10
0.5
1
1.5
2
∂f(x,y)∂x
= 2x.= 0 ⇒ x = 0,
∂f(x,y)∂y
= 2y.= 0 ⇒ y = 0.
Since we have only one critical point, it is either the maximum or the
minimum. We observe that f(x, y) takes only nonnegative values.
Thus, we see that the origin is the minimum point.
Minima, Maxima, Saddle points – p.2/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Find the extreme points of
f(x, y) = xy − x2 − y2 − 2x − 2y + 4.
Minima, Maxima, Saddle points – p.3/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Find the extreme points of
f(x, y) = xy − x2 − y2 − 2x − 2y + 4.
The function is differentiable and has no boundary points.
Minima, Maxima, Saddle points – p.3/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Find the extreme points of
f(x, y) = xy − x2 − y2 − 2x − 2y + 4.
fx = ∂f(x,y)∂x
= y − 2x − 2, fy = ∂f(x,y)∂y
= x − 2y − 2.
Thus, x = y = −2 is the critical point.
Minima, Maxima, Saddle points – p.3/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Find the extreme points of
f(x, y) = xy − x2 − y2 − 2x − 2y + 4.
fx = ∂f(x,y)∂x
= y − 2x − 2, fy = ∂f(x,y)∂y
= x − 2y − 2.
Thus, x = y = −2 is the critical point.
fxx = ∂2f(x,y)∂x2 = −2 = ∂2f(x,y)
∂y2 = fyy, fxy = ∂2f(x,y)∂x∂y
= 1.
Minima, Maxima, Saddle points – p.3/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Example . Find the extreme points of
f(x, y) = xy − x2 − y2 − 2x − 2y + 4.
fx = ∂f(x,y)∂x
= y − 2x − 2, fy = ∂f(x,y)∂y
= x − 2y − 2.
Thus, x = y = −2 is the critical point.
fxx = ∂2f(x,y)∂x2 = −2 = ∂2f(x,y)
∂y2 = fyy, fxy = ∂2f(x,y)∂x∂y
= 1.
The discriminant (Jacobian) of f at (a, b) = (−2,−2) is
fxx fxy
fxy fyy
= fxxfyy − f2xy = 4 − 1 = 3.
Since fxx < 0, fxxfyy − f2xy > 0 ⇒ f has a local maximum at (−2,−2).
Minima, Maxima, Saddle points – p.3/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Theorem . The extreme values for f(x, y) can occur only at
i. Boundary points of the domain of f .
ii. Critical points (interior points where fx = fy = 0, or points where
fx or fy fails to exist).
Minima, Maxima, Saddle points – p.4/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Scalar Functions
Theorem . If the first and second order partial derivatives of f are
continuous throughout an open region containing a point (a, b) and
fx(a, b) = fy(a, b) = 0, you may be able to classify (a, b) with the
second derivative test:
i. fxx < 0, fxxfyy − f 2xy > 0 at (a, b) ⇒ local maximum;
ii. fxx > 0, fxxfyy − f 2xy > 0 at (a, b) ⇒ local minimum;
iii. fxxfyy − f 2xy < 0 at (a, b) ⇒ saddle point;
iv. fxxfyy − f 2xy = 0 at (a, b) ⇒ test is inconclusive (f is singular).
Minima, Maxima, Saddle points – p.4/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Definition . The quadratic termf(x, y) = ax2 + 2bxy + cy2
is positive definite (negative definite) if and only if a > 0(a < 0) and ac − b2 > 0.
Minima, Maxima, Saddle points – p.5/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Definition . The quadratic termf(x, y) = ax2 + 2bxy + cy2
is positive definite (negative definite) if and only if a > 0(a < 0) and ac − b2 > 0.f has a minimum (maximum) atx = y = 0 if and only if fxx(0, 0) > 0 (fxx(0, 0) < 0)
and fxx(0, 0)fyy(0, 0) > f 2xy(0, 0).
Minima, Maxima, Saddle points – p.5/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Definition . The quadratic termf(x, y) = ax2 + 2bxy + cy2
is positive definite (negative definite) if and only if a > 0(a < 0) and ac − b2 > 0.f has a minimum (maximum) atx = y = 0 if and only if fxx(0, 0) > 0 (fxx(0, 0) < 0)
and fxx(0, 0)fyy(0, 0) > f 2xy(0, 0).
If f(0, 0) = 0, we term f as positive (negative)semi-definite provided the above conditions hold.
Minima, Maxima, Saddle points – p.5/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Now, we are able to introduce matrices to the quadraticforms:
ax2 + 2bxy + cy2 = [x, y]
a b
b c
x
y
.
Minima, Maxima, Saddle points – p.6/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a purequadratic form: it has a stationary point at the origin and nohigher terms.
Minima, Maxima, Saddle points – p.6/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a purequadratic form: it has a stationary point at the origin and nohigher terms.
xAT x = [x1, x2, · · · , xn]
a11 a12 · · · a1n
a21 a22 · · · a2n
......
. . ....
an1 an2 · · · ann
x1
x2
...
xn
Minima, Maxima, Saddle points – p.6/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a purequadratic form: it has a stationary point at the origin and nohigher terms.
xAT x = [x1, x2, · · · , xn]
a11 a12 · · · a1n
a21 a22 · · · a2n
......
. . ....
an1 an2 · · · ann
x1
x2
...
xn
= a11x21 + a12x1x2 + · · · + annx
2n =
∑n
i=1
∑n
j=1 aijxixj.
Minima, Maxima, Saddle points – p.6/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Definition . If A is such that aij = ∂2f∂xi∂xj
(hence
symmetric), it is called the Hessian matrix.
Minima, Maxima, Saddle points – p.7/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Definition . If A is such that aij = ∂2f∂xi∂xj
(hence
symmetric), it is called the Hessian matrix.If A is positive definite (xTAx > 0, ∀x 6= θ) and if f hasa stationary point at the origin (all first derivatives at theorigin are zero), then f has a minimum.
Minima, Maxima, Saddle points – p.7/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Remark . Let f : Rn 7→ R and x∗ ∈ R
n be the local minimum,
∇f(x∗) = θ and ∇2f(x∗) is positive definite.
Minima, Maxima, Saddle points – p.8/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Remark . Let f : Rn 7→ R and x∗ ∈ R
n be the local minimum,
∇f(x∗) = θ and ∇2f(x∗) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where ‖∆x‖ is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive.
Minima, Maxima, Saddle points – p.8/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Remark . Let f : Rn 7→ R and x∗ ∈ R
n be the local minimum,
∇f(x∗) = θ and ∇2f(x∗) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where ‖∆x‖ is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive. Then,
f(x∗ + ∆x) ∼= f(x∗) + ∆xT∇f(x∗) + 12∆xT∇2f(x∗)∆x.
Minima, Maxima, Saddle points – p.8/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Quadratic forms
Remark . Let f : Rn 7→ R and x∗ ∈ R
n be the local minimum,
∇f(x∗) = θ and ∇2f(x∗) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where ‖∆x‖ is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive. Then,
f(x∗ + ∆x) ∼= f(x∗) + ∆xT∇f(x∗) + 12∆xT∇2f(x∗)∆x.
The second term is zero since x∗ is a critical point and the third term is
positive since the Hessian evaluated at x∗ is positive definite. Thus, the
left hand side is always strictly greater than the right hand side,
indicating the local minimality of x∗.
Minima, Maxima, Saddle points – p.8/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let f(x1, x2) = 13x3
1 + 12x2
1 + 2x1x2 + 12x2
2 − x2 + 19. Find the stationaryand boundary points, then find the minimizer and the maximizer over−4 ≤ x2 ≤ 0 ≤ x1 ≤ 3
24 35 24 3524 35 24 35 24 35 24 3524 35 24 35 24 35 24 35
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let f(x1, x2) = 13x3
1 + 12x2
1 + 2x1x2 + 12x2
2 − x2 + 19. Find the stationaryand boundary points, then find the minimizer and the maximizer over−4 ≤ x2 ≤ 0 ≤ x1 ≤ 3
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
24 35 24 3524 35 24 35 24 35 24 3524 35 24 35 24 35 24 35
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let f(x1, x2) = 13x3
1 + 12x2
1 + 2x1x2 + 12x2
2 − x2 + 19. Find the stationaryand boundary points, then find the minimizer and the maximizer over−4 ≤ x2 ≤ 0 ≤ x1 ≤ 3
∇f(x) =
∂f∂x1
∂f∂x2
=
x21 + x1 + 2x2
2x1 + x2 − 1
.=
0
0
⇒
(x1 − 1)(x1 − 2) = 0
x2 = 1 − 2x1
24 35 24 3524 35 24 35 24 35 24 3524 35 24 35 24 35 24 35
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
∇f(x) =
∂f∂x1
∂f∂x2
=
x21 + x1 + 2x2
2x1 + x2 − 1
.=
0
0
⇒
(x1 − 1)(x1 − 2) = 0
x2 = 1 − 2x1
Therefore, xA =
24 1
−1
35 , xB =
24 2
−3
35 are stationary points inside the region
defined by −4 ≤ x2 ≤ 0 ≤ x1 ≤ 3.
24 35 24 35 24 35 24 3524 35 24 35 24 35 24 35
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
∇f(x) =
∂f∂x1
∂f∂x2
=
x21 + x1 + 2x2
2x1 + x2 − 1
.=
0
0
⇒
(x1 − 1)(x1 − 2) = 0
x2 = 1 − 2x1
Therefore, xA =
24 1
−1
35 , xB =
24 2
−3
35 are stationary points inside the region
defined by −4 ≤ x2 ≤ 0 ≤ x1 ≤ 3. Moreover, we have the following boundaries
xI =
24 0
x2
35 , xII =
24 3
x235 and xIII =
24 x1
−4
35 , xIV =
24 x1
0
35
defined by
xC =
24 0
035 , xD =
24 0
−4
35 , xE =
24 3
0
35 , xF =
24 3
−4
35.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let the Hessian matrix be
∇2f(x) =
∂2f∂x1∂x1
∂2f∂x1∂x2
∂2f∂x2∂x1
∂2f∂x2∂x2
=
2x1 + 1 2
2 1
.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let the Hessian matrix be
∇2f(x) =
∂2f∂x1∂x1
∂2f∂x1∂x2
∂2f∂x2∂x1
∂2f∂x2∂x2
=
2x1 + 1 2
2 1
. Then, we
have ∇2f(xA) =
3 2
2 1
and ∇2f(xB) =
5 2
2 1
.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let the Hessian matrix be
∇2f(x) =
∂2f∂x1∂x1
∂2f∂x1∂x2
∂2f∂x2∂x1
∂2f∂x2∂x2
=
2x1 + 1 2
2 1
. Then, we
have ∇2f(xA) =
3 2
2 1
and ∇2f(xB) =
5 2
2 1
.
Let us check the positive definiteness of ∇2f(xA):
vT∇2f(xA)v = [v1, v2]24 3 2
2 1
3524 v1
v2
35 = 3v2
1+ 4v1v2 + v2
2.
If v1 = −0.5 and v2 = 1.0, we will have vT∇2f(xA)v < 0. On the other hand, if
v1 = 1.5 and v2 = 1.0, we will have vT∇2f(xA)v > 0. Thus, ∇2f(xA) is indefinite.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let the Hessian matrix be
∇2f(x) =
∂2f∂x1∂x1
∂2f∂x1∂x2
∂2f∂x2∂x1
∂2f∂x2∂x2
=
2x1 + 1 2
2 1
. Then, we
have ∇2f(xA) =
3 2
2 1
and ∇2f(xB) =
5 2
2 1
.
Let us check ∇2f(xB):
vT∇2f(xB)v = [v1, v2]
24 5 2
2 13524 v1
v2
35 = 5v2
1+ 4v1v2 + v2
2= v2
1+ (2v1 + v2)2 > 0.
Thus, ∇2f(xB) is positive definite and xB =
24 2
−3
35 is a local minimizer with
f(xB) = 19.166667Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let us check the boundary defined by xI :
f(0, x2) =1
2x2
2 − x2 + 19 ⇒df(0, x2)
dx2= x2 − 1
.= 0 ⇒ x2 = 1.
Since d2f(0,x2)dx2
2
= 1 > 0, x2 = 1 > 0 is the local minimizer outside
the feasible region. As the first derivative is negative for
−4 ≤ x2 ≤ 0, we will check x2 = 0 for minimizer and x2 = −4 for
maximizer.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let us check the boundary defined by xII :
f(3, x2) =1
2x2
2 + 5x2 +65
2⇒
df(3, x2)
dx2= x2 + 5
.= 0 ⇒ x2 = −5.
Since d2f(0,x2)dx2
2
= 1 > 0, x2 = −5 < −4 is the local minimizer
outside the feasible region. As the first derivative is positive for
−4 ≤ x2 ≤ 0, we will check x2 = −4 for minimizer and x2 = 0 for
maximizer.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let us check the boundary defined by xIII :
f(x1, 0) =1
3x3
1+1
2x2
1+19 ⇒df(x1, 0)
dx1= x2
1+x1.= 0 ⇒ x1 = 0,−1.
Since d2f(x1,0)dx2
1
= 2x1 + 1, x1 = 0 is the local minimizer
(d2f(0,0)dx2
1
= 1 > 0) on the boundary, and x1 = −1 is the local
maximizer (d2f(−1,0)dx2
1
= −1 < 0) outside the feasible region. As
the first derivative is positive for 0 ≤ x2 ≤ 3, we will check x2 = 3
for maximizer.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
Let us check the boundary defined by xIV :
f(x1,−4) = 13x3
1 + 12x2
1 − 8x1 + 31 ⇒ df(x1,−4)dx1
= x21 + x1 − 8
.= 0
⇒ x1 = −1±√
1+322 . Since d2f(x1,−4)
dx2
1
= 2x1 + 1 again, the positive
root x1 = −1+√
332 = 2.3723 is the local minimizer
(d2f(2.3723,0)dx2
1
> 0), and the negative root is the local maximizer but
it is outside the feasible region. As the first derivative is positive
for 0 ≤ x2 ≤ 3, we will check x2 = 3 for maximizer again.
Minima, Maxima, Saddle points – p.9/9
c©Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Collaborative Work:
0
0,2
0,4
0,6
0,8 1
1,2
1,4
1,6
1,8 2
2,2
2,4
2,6
2,8 3
-4
-3,5
-2,9
-2,4
-1,9
-1,4
-0.9
-0.4
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
B
C
E
To sum up, we have to consider (2,−3), (0, 0) and (2.3723,−4)
for the minimizer; (3, 0) and (0,−4) for the maximizer:
f(2,−3) = 19.16667, f(0, 0) = 19, f(2.3723,−4) = 19.28529
⇒ (0, 0) is the minimizer!
f(3, 0) = 32.5, f(0,−4) = 31 ⇒ (3, 0) is the maximizer!
Minima, Maxima, Saddle points – p.9/9