Numerical Methods and Optimization

Unconstrained Optimization 347

1: Choose ¯α>0,ρ,c ∈ (0, 1)

2: α ← ¯α

3: repeat

4: α ← ρα

5: until f(x

(k)

+ αd

(k)

) ≤ f(x

(k)

)+cα∇f(x

(k)

)

T

d

(k)

6: return α

k

= α

Exercises

13.1. A company manufactures two similar products. The manufacturing cost

is $40 for a unit of product 1 and $42 for a unit of product 2. Assume

that the company can sell q

1

= 150 − 2p

1

+ p

2

units of product 1 and

q

2

= 120 + p

1

− 3p

2

units of product 2, where p

1

and p

2

are prices

charged for product 1 and product 2, respectively. The company’s goal

is to maximize the total profit. What price should be charged for each

product? How many units of each product should be produced? What

is the optimal profit? ...