APM 462 Nonlinear Optimization Summer 2025
Assignment 5
Due date: July 26, 2025; 11.59 pm
Problems named TUT are for discussion during the tutorials. Problems named
HW are to be handed in on Crowdmark until the due date. Only selected HW
problems will be graded.
LY abbreviates Luenberger and Ye, H abbreviates Hendrix (Introduction to
Nonlinear and Global Optimization 2010), vB abbreviates van Brunt, BV ab-
breviates Boyd and Vandenberghe (Convex Optimization 2009), S abbreviates
Santambrogio (A Course in the Calculus of Variations 2023).
TUT 1 Consider the functional
J(y) =
∫
Ω
1
2
|∇y|2 + yg dx
where Ω ⊂ Rn is a domain, g : Ω→ R and y : Ω→ R are functions.
Show that J is convex. Find the Euler-Lagrange equation for J .
TUT 2 (vB 10.2 1.) Suppose that J is a functional of the form
J(y) =
∫ x1
x0
f(x, y, y′, y′′) dx
where f ∈ C∞([x0, x1]×R×R×R). Derive an expression for the second
variation of J .
TUT 3 (vB 10.4 3.) Let
J(y) =
∫ x1
x0
f(x, y′) dx
with f ∈ C∞([x0, x1] × R) and assume that f satisfies the strengthened
Legendre condition along an extremal y. Prove that there are no points
conjugate to x0. Show, using elementary arguments, that δ
2J(η, y) ≥ 0
for all η ∈ H. Derive the Jacobi accessory equation and show by solving
this equation that any nontrivial solution y can have at most one zero.
TUT 4 (S 1.8 1.9) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C2([0, 1]) : y(0) = a, y(1) = b}
1
where a, b ∈ R fixed and
J(y) =
∫ 1
0
L(x, y, y′) dx
with L ∈ C2([0, 1]×R×R). Assume that y is a minimizer, show that we
have
∂L
∂y′
(x, y(x), y′(x)) ≥ 0
for all x ∈ [0, 1].
TUT 5 (S 1.8 1.10) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C1([0, 1]) : y(0) = a, y(1) = b}
where a, b ∈ R fixed and
J(y) =
∫ 1
0
(
(y′)2 − 1)2 + f(y) dx
with f ∈ C2(R). Show that if |a − b| ≤ 1√
3
then the problem does not
admit a solution.
TUT 6 (S 1.8 1.11) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C1([0, 1])}
with
J(y) =
∫ 1
0
(y′)2 + arctan(y) dx.
Show that the problem does not admit a solution. Show that a solution
exists if we add the boundary condition y(0) = 0. Write down the Euler-
Lagrange equation for this problem and prove that every solution is in
C∞([0, 1]).
2
HW 1 (vB 10.3 2.) Let
J(y) =
∫ x1
x0
1 + y2
(y′)2
dx
and suppose that J has a local extremum at y. Use the Legendre condition
to determine the nature of the extremum.
HW 2 (vB 10.4 2.) Let
J(y) =
∫ x1
x0
(y′)2 − y′y + y2 dx.
Show, using elementary arguments, that δ2J(η, y) ≥ 0 for all η ∈ H.
Derive the Jacobi accessory equation and show by solving this equation
that any nontrivial solution u to the Jacobi accessory equation can have
at most one zero.
HW 3 (vB 10.7 3.) Show that the function f(y, y′) =
√
y2 + (y′)2 is convex on R2 and that
the function g(y, y′) = ey
√
1 + (y′)2 is convex on Ω := {(y, y′) ∈ R2 :
|y′| ≤ 1}.
HW 4 (S 1.8 1.5) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C2([0, T ]) : y(0) = 1}
where
J(y) =
∫ T
0
e−x
(
(y′)2 + 5y2
)
dx
for T > 0.
(a) Find the minimizer for this problem (and prove that it is a mini-
mizer).
(b) Is the minimizer unique?
(c) Compute the minimal value of the problem. What happens if T →
+∞?
HW 5 (S 1.8 1.15) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C2([0, 2π]) : y(0) = 0, y(2π) = 1}
where
J(y) =
∫ 2π
0
|y′(x)|2
2 + sin(x)
dx
Find the minimizer (and prove that it is a minimizer) and the minimal
value of J .
3
HW 6 (S 1.8 1.4) Consider the problem
min J(y)
s.t. y ∈ S = {y ∈ C1([0, 1]) : y(0) = 0}
where
J(y) =
∫ 1
0
1
2
y′2 + xy(x) +
1
2
y2 dx
Find the minimizer(s) and the minimal value of J , proving that the min-
imizers are actually minimizers.
4

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