THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MMAT5240 Optimization and Modeling (Spring 2020)
Homework 1
Due Date: Feb. 5, 2020
1. (a) Estimate the error in approximating
√
1 + x by 1 + x
2
when 0 < x < 0.01.
(b) For what positive values of x can you replace ln(1 + x) by x with an error of
magnitude no greater than 1% of the value of x?
2. Define the functions f : R2 → R and g : R→ R2 by
f(x1, x2) =
x21
6
+
x22
4
, g(t) = [3t + 5, 2t− 6]T
Let F : R→ R be given by F (t) = f(g(t)). Evaluate dF
dt
(t) using the chain rule.
3. Consider the following function f : R2 → R:
f(x) = xT
[
2 5
−1 1
]
x + xT
[
3
4
]
+ 7
(a) Find the directional derivative of f at [0, 1]T in the direction [1, 0]T .
(b) Find all points that satisfy the first-order necessary condition for f . Does f
have a minimizer? If it does, then find all minimizer(s); otherwise, explain why
it does not.
4. Consider the problem
minimize f(x)
subject to x ∈ Ω
where x = [x1, x2]
T , f : R2 → R is given by f(x) = 4x21 − x22, and Ω = {x :
x21 + 2x1 − x2 ≥ 0, x1 ≥ 0, x2 ≥ 0}.
(a) Does the point x∗ = 0 = [0, 0]T satisfy the first-order necessary condition?
(b) Does the point x∗ = 0 satisfy the second-order necessary condition?
(c) Is the point x∗ = 0 a local minimizer of the given problem?
5. Consider the problem
maximize c1x1 + c2x2
subject to x1 + x2 ≤ 1
x1, x2 ≥ 0,
where c1, c2 are constants such that c1 > c2 ≥ 0. This is a linear programming
problem. Assuming that the problem has an optimal feasible solution, use the first-
order necessary condition to show that the unique optimal feasible solution x∗ is
[1, 0]T .
Hint: First show that x∗ cannot lie in the interior of the constraint set. Then,
show that x∗ cannot lie on the line segments L1 = {x : x1 = 0, 0 ≤ x2 < 1},
L2 = {x : x2 = 0, 0 ≤ x1 < 1}, L3 = {x : x2 = 1− x1, 0 ≤ x1 < 1}.