APM462: Homework 3
Due: Sat June 15 before 11:59pm
Submit your solutions to the homework on Crowdmark.
(1) Given a positive number a, write it as the product of n positive numbers
xi, such that the sum of the inverses of xi is as small as possible.
(2) Let n ≥ 1 and x, y ≥ 0. Prove the following inequality:(
x+ y
2
)n
≤ x
n + yn
2
.
Hint: Set x+ y = a and minimise the right hand side.
(3) Let (x0, y0) be a minimiser for the following constrained minimisation
problem:
minimise: F (x, y)
subject to: H(x, y) = y − h(x) = 0,
(a) Convert this problem into an unconstrained minimisation problem of
the form:
minimise: f(x)
subject to: x ∈ R,
Hint: Use the constraint.
(b) Find the first order necessary conditions that a minimiser x0 of f(x)
must satisfy.
(c) Use part (b) to show that ∇F (x0, y0) is a multiple of ∇H(x0, y0).
(4) Let Q be a positive definite symmetric 2 × 2 matrix with eigenvalues
0 < λ1 ≤ λ2. Consider the problem
maximise: f(x) = x
TQx
(xT x)2
subject to: |x|2 ≥ 1,
where x ∈ Rn.
(a) Show that the above problem is equivalent to
maximise: f(x) = xTQx
subject to: |x|2 = 1.
Hint: You may want to start by showing that f(ax) = f(x)a2 for
0 ̸= a ∈ R.
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(b) Use Lagrange multipliers to find the candidates for maximisers of
the problem in (a). Express the maximum value in terms of the
eigenvalues of Q.
(c) What do the second order conditions say bout the candidates you
found in part (b)?
(5) Consider the problem
minimise: f = x2 + 9y2
subject to: g1 = 2− x− 9y ≤ 0
g2 = 1− x− y ≤ 0.
(a) Classify all the feasible points as regular or irregular.
(b) Use the KKT conditions to find the candidate for a local minimiser.
You may find it helpful to draw a diagram.
(c) Are the second order conditions satisfied at the point you found in
part (b)?
(6) Consider the problem
minimise: f = −2x+ 3y2
subject to: g1 = (x− 1)2 + y2 > 1
g2 = (x− 1)2 + y2 ≤ 4
g3 = x ≥ 0.
Hint: Note the strict inequality.
(a) Classify all the feasible points as regular or irregular.
(b) Find the points which satisfy the first order necessary conditions.
(c) Are the second order conditions satisfied at the points you found in
part (b)?
(d) What is the global minimum?
(7) Fix αi ∈ R and consider the minimisation problem
minimise: f = −∑ni=1 log(αi + xi)
subject to: x1, . . . , xn ≥ 0
x1 + · · ·+ xn = 1.
(a) Using the first order conditions, show that xi has the form:
xi = max
{
0,
1
λ
− αi
}
for some λ ∈ R.
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(b) Argue that λ is unique and, therefore, the solution to the minimisation
problem is unique. You do not need to solve for λ.
The following 7 problems—the ones indexed by Roman numerals—are
for practice only and are not to be turned in.
I. During lecture, we showed that (x0, y0, z0) = (l, l, l) is a strict local min-
imiser for the problem
maximise: xyz
subject to: 2xy + 2xz + 2yz = A
x, y, z > 0.
Show that (l, l, l) is the global minimiser.
II. Let f : Rn → R be a C1 function. Recall that the graph of f is the surface
M = {(x, f(x)) ∈ Rn × R}. Given a point p = (x0, f(x0)) ∈ M on the
graph of f , find a formula for the tangent space TpM in terms of ∇f(x0).
Hint: you can think of M as the level set h(x, z) = f(x)− z = 0.
III. Consider the following problem where f : Rn → R and gi : Rn → R,
1 ≤ i ≤ n, are C1
minimise: f(x)
subject to: gi ≤ 0 for 1 ≤ i ≤ n.
Consider modifying the problem by replacing f and each gi with their first
order Taylor approximation. Show that x∗ is an optimal solution to the
original problem iff it is an optimal solution to the modified problem.
IV. Let a, b > 0 with a ≤ b. Find n numbers x1, . . . , xn between a and b such
that the function
f =
x1 · · ·xn
(a+ x1)(x1 + x2) · · · (xn−1 + xn)(xn + b)
is maximised.
V. The distance between two closed convex sets C and D in Rn is defined as
min{|x, y|∣∣x ∈ C, y ∈ D}}
Suppose that C is the closed ball of radius 1 centred at (−1, 1) and D is
everything including and below the parabola x2 = −x21. Express this as
an optimisation problem of the form
minimise: f(x1, x2)
subject to: gi(x1, x2) ≤ 0.
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You do not need to solve the resulting problem(but feel free to try).
VI. Let a ∈ R be constant. Consider the problem
minimise: ax− 4y − 2z
subject to: x2 + y2 ≤ 2
x2 + z2 ≤ 2
y2 + z2 ≤ 2.
Find all, if any, values of a which make (x, y, z) = (1, 1, 1) an optimal
solution to the problem.
VII. Let a be a number. Find the smallest possible value of each variable for
which the following equation holds:
(x2 + y2 + z2)2 = a2(x2 + y2 − z2).