Prof Dr .J Fliege
Mathematical Sciences
University of Southampton
MATH6184-Nonlinear Optimization
Exercise Sheet 2
You are not required to hand in the answers to these problems.
1. Consider the function f (x) = x3 − 3x2 + 11x at the point x = 3.
(a) Derive the approximation to f (x + td) as on slide 14 of Section 2.
(b) Derive the approximation to f (x + td) as on slide 44 of Section 2.
(c) Plot the original function and both approximations in the vicinity of x. How accu-
rate to the approximations appear to be? Which is better?
2. Do exercise 1 (above) for the function f (x) = 18x − 50 ln(x) at x = 16.
3. Consider the function f (x1 , x2 ) = x31 −5x1 x2 +6x22 at the point x = (0, 2)T and direction
d = (1, −1)T .
(a) Derive the approximation to f (x + td) as on slide 14 of Section 2.
(b) Derive the approximation to f (x + td) as on slide 44 of Section 2.
(c) Plot the original function and both approximations as functions of t. How accurate
to the approximations appear to be? Which is better?
4. Do exercise 3 (above) for the function f (x1 , x2 ) = 13x1 − 6x1 x2 + 8/x2 , x = (2, 1)T and
d = (3, 1)T .
5. A site for a new service facility has been chosen. It is now necessary to determine the
size of the facility. The facility is to be located in the center of the circular area that
it serves, and the radius r is the only decision variable. The minimum radius of the
circular area that the facility is supposed to serve is 10km, while the maximum radius
is 100km. Service calls occur with a uniform density of 50 calls/km2 in the area. The
operating cost of the facility are as follows: 2 million £ fixed cost and 0.3 million £ per
km2 covered. Moreover, transportation costs per call are 5£ per kilometer travelled to
the customer. Travel takes place along a straight line from the center of the circle served
to the customer.
Formulate an unconstrained nonlinear programming problem with one variable r to
choose a market area radius that minimizes the average total cost per service call.
Evaluate any calculus integrals in your objective.
6. (This exercise considers some additional computational tools and is therefore non-
examinable.)
Consider the unconstrained nonlinear programming problem
max x1 x2 − 5(x1 − 2)4 − 3(x2 − 5)4 .
1
Mathematical Sciences
University of Southampton
MATH6184-Nonlinear Optimization
Exercise Sheet 2
You are not required to hand in the answers to these problems.
1. Consider the function f (x) = x3 − 3x2 + 11x at the point x = 3.
(a) Derive the approximation to f (x + td) as on slide 14 of Section 2.
(b) Derive the approximation to f (x + td) as on slide 44 of Section 2.
(c) Plot the original function and both approximations in the vicinity of x. How accu-
rate to the approximations appear to be? Which is better?
2. Do exercise 1 (above) for the function f (x) = 18x − 50 ln(x) at x = 16.
3. Consider the function f (x1 , x2 ) = x31 −5x1 x2 +6x22 at the point x = (0, 2)T and direction
d = (1, −1)T .
(a) Derive the approximation to f (x + td) as on slide 14 of Section 2.
(b) Derive the approximation to f (x + td) as on slide 44 of Section 2.
(c) Plot the original function and both approximations as functions of t. How accurate
to the approximations appear to be? Which is better?
4. Do exercise 3 (above) for the function f (x1 , x2 ) = 13x1 − 6x1 x2 + 8/x2 , x = (2, 1)T and
d = (3, 1)T .
5. A site for a new service facility has been chosen. It is now necessary to determine the
size of the facility. The facility is to be located in the center of the circular area that
it serves, and the radius r is the only decision variable. The minimum radius of the
circular area that the facility is supposed to serve is 10km, while the maximum radius
is 100km. Service calls occur with a uniform density of 50 calls/km2 in the area. The
operating cost of the facility are as follows: 2 million £ fixed cost and 0.3 million £ per
km2 covered. Moreover, transportation costs per call are 5£ per kilometer travelled to
the customer. Travel takes place along a straight line from the center of the circle served
to the customer.
Formulate an unconstrained nonlinear programming problem with one variable r to
choose a market area radius that minimizes the average total cost per service call.
Evaluate any calculus integrals in your objective.
6. (This exercise considers some additional computational tools and is therefore non-
examinable.)
Consider the unconstrained nonlinear programming problem
max x1 x2 − 5(x1 − 2)4 − 3(x2 − 5)4 .
1
