• For NLPs having multiple local optimal

solutions, the Solver may fail to find the

optimal solution because it may pick a local

extremum that is not a global extremum.

22

Example 2

• Trucko is trying to determine where they should locate a single warehouse. The positions in the x‐y plane of their four customers and the number of shipments made annually to each customer are given. Trucko wants to locate the warehouse to minimize the total distance trucks must travel annually from the warehouse to the four customers. Customer

X‐coordinate

Y‐coordinate

Number of

Shipments

1

5

10

200

2

10

5

150

3

0

12

200

4

12

0

300

23

1

Example 3

• Q& H Company advertises on soap operas and football games. Each soap opera ad costs $50000, and each football game ad costs $10000. Giving all figures in millions of viewers, S soap opera ads are bought, they will be seen by 5S men and 20 S women. If F football Ads are bought they will be seen by 17F men and 7F women. Q&H wants at least40 million men and at least 60 million women to see its ads.

• Formulate the NLP

• Suppose the number of women reached by F football ads and S soap opera ads is 7F+ 20 S ‐0.2 (FS). Why might this be a more realistic representation of the number of women viewers seeing Q&H’s ads?

24

Example 3: cont’d

• Let S = soap opera ads and

F = football ads. Then we wish to

min z = 50S + 100F

st 5S1/2 + 17F1/240 (men)

20S1/2 + 7F1/260 (women)

S0, F0

25

2

Review of Differential Calculus

• The equation

lim f ( x) c

x a

means that as x gets closer to a (but not equal to a), the value of f(x) gets arbitrarily close to c.

• A function f(x) is continuous at a point if

lim f ( x) f (a )

xa

If f(x) is not continuous at x=a, we say that f(x) is discontinuous (or has a discontinuity) at a.

26

Review of Differential Calculus

• The derivative of a function f(x) at x = a

(written f’(a)] is defined to be

lim

x 0

f (a x) f (a)

x

• The partial derivative of f(x1, x2,…,xn) with

respect to the variable xi is written

f

, where

xi

f ( x1 ,..., xi xi ,..., xn ) f ( x1 ,..., xi ,...xn ) f

lim

xi xi 0

xi

27

3

Review of Differential Calculus

• Suppose that for each i, we increase xi by a small amount Δxi. Then the value of f will increase by

approximately

in

f

x xi

i

1

i

• We will also use second‐order partial derivatives extensively. We use the notation

2

xi x j

to denote a second‐order partial derivative.

28

Review of Differential Calculus

• The demand f(p,a)=30000p‐2a1/6 for a product

depends on p=price and a=dollars spent

advertising the product.

• Is demand an increasing or decreasing function of price?

• Is demand an increasing or decreasing function of advertising expenditure?

• If p=10 and a= 1000000, by how much

approximately will a $1 cut in price increase the

demand?

29

4

Review of Differential Calculus

• n‐th order Taylor series expansion

f (i ) (a) i f ( n 1) ( p ) n 1

h

h

i!

(n 1)!

i 1

where h 0 for some p between a and a h

in

f ( a h) f ( a )

Example: Find first order Taylor expansion for

f ( h) e h e 0

f ( x) e x

at x=0

e0 h e p h 2

e p h2

1 h

1!

2!

2!

30

Convex and Concave Functions

• A function f(x1, x2,…,xn) is a convex function on a convex set S if for any x’ S and x’’ S

f [cx’+(1‐ c)x’’] ≤ cf(x’)+(1‐c)f(x’’)

holds for 0 ≤ c ≤ 1.

6

5

4

3

2

1

0

‐1

0

1

2

3

4

5

31

5

Convex and Concave Functions

• A function f(x1, x2,…,xn) is a concave function on a convex set S if for any x’ S and x’’ S f [cx’+(1‐ c)x’’] ≥ cf(x’)+(1‐c)f(x’’)

holds for 0 ≤ c ≤ 1.

0

‐1

0

1

2

3

4

5

‐1...