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ANALYSIS AND OPTIMIZATION, PROBLEM SET 6
Due Monday 3/30, 5pm
Problem 1.
(1) Consider the subset S ⊂ R3 defined by
x4 + 2xy + y 4 + yz + z 3 = 2
Show there is a C 1 function g : R2 → R defined near (1, 1) such that
for (x, y, z) near (1, 1, −1), then (x, y, z) ∈ S if and only if z = g(x, y).
Compute ∇g near at (1, 1).
(2) Consider the set S ⊂ R3 of points (x, y, z) such that
x2 + xy + xz + y 2 + yz + z 2 − 25 = 0, 36x2 + 9y 2 + 4z 2 − 108 = 0.
Show that the point (1, 2, 3) is a regular point of S.
Problem 2. Consider a chain of n springs, with lengths `1 , `2 , . . . , `n and (fixed)
stiffnesses k1 , . . . , kn . The energy of the system for a given set of spring lengths is
n
1X 2
ki `i .
E=
2 i=1
Find the configuration of lengths which minimizes the energy under the constraint
`1 + `2 + · · · + `n = L, where L is fixed. (Assume that such a minimum exists; when
we discuss convex constrained optimization, we will see why it must).
Problem 3. (SHSS, 1.8 problem 3): Use bordered Hessians to show the quadratic
form −5x2 + 2xy + 4xz − y 2 − 2z 2 is negative definite on the subspace of R3 defined
by x + y + z = 0, 4x − 2y + z = 0.
Problem 4. Consider a closed box with sides x, y, z and fixed volume V . The goal
is to minimize the surface area of the box for fixed V .
(1) Write the equation of the constraint set. Are there any points that are not
regular?
(2) Set up the Lagrange multiplier problem and write down the necessary first
order conditions.
(3) Solve the minimization problem. Is your minimizer regular?
(4) Check the sufficient conditions for second order minima - are they satisfied
here?
Problem 5.
(1) Maximize and minimze the function x2 + y 2 subject to the
constraint 4x2 + 2y 2 = 4. Use Lagrange multipliers to find all solutions to
the first order criteria. Check in which cases the second order conditions
for a minimizer and maximizer are satisfied. For this part, do not use
bordered Hessians; it is easier to check directly that Hess(f ) − λHess(h) is
positive or negative definite on the constraint.
(2) Consider the optimization problem for the function x2 + y 2 + z 2 subject
to the constraint x + y + z = 1. Use Lagrange multipliers to find the first
order criteria for a minimizer. Show that the second-order condition for a
local minimum are satisfied (using whichever approach you want).
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ANALYSIS AND OPTIMIZATION, PROBLEM SET 6
Problem 6. Find the global minimum and maximum of the function f (x, y) =
2x3 + 4y 3 on the set S = {x2 + y 2 ≤ 1}. (The maximum theorem tells us that
there is a global max/min on S. If the min or max occurs in the interior of S, we
can find it by looking at the gradient of f . If the min/max doesn’t occur in the
interior, we can find it using Lagrange multipliers on the unit circle. Write down
all possibilities and compare values of f .)
Problem 7. Let A be a n × n symmetric matrix. Recall the spectral theorem
states that A has an orthonormal basis of eigenvectors. The key step in the proof
of the spectral theorem is to show that there is at least one eigenvector (there are
matrices with no real eigenvectors, so this is already not obvious!)
(1) Let S = {x ∈ Rn |kxk = 1} denote the unit sphere and let Q(x) = xtr Ax
denote the quadratic form associated to A. Show that Q restricted to S
has a maximum value and a minimum value realized somewhere on S.
(2) Let g(x) = kxk2 so that S = {x|g(x) = 1}. Show that ∇g(x) = 2xtr .
(3) Show that if x0 ∈ S is a point where Q achieves its global minimum on S,
then x0 is an eigenvector for A. What is the corresponding eigenvalue?