APM462: Homework 1
Due: Sat May 18 before 11:59pm
Submit your solutions to the homework on Crowdmark.
The problems labelled [J] are from the notes by Prof. Jerrard for Mat237.
(1) [J] Chapter 1.2: Less basic problems: 3.
(2) [J] Chapter 1.3: Less basic problem: 3.
(3) [J] Chapter 1.4: Less basic problems: 6 (give two different proofs using
the two different characterizations of compact sets in Rn).
(4) [J] Chapter 3.1: Other questions: 1. Hint: for the second part use the
quadratic formula to solve for y in terms of x
(5) (a) Let f(x) = b ·Ax, where A is an n×m matrix, x ∈ Rm, and b ∈ Rn
Show that ∇f(x) = AT b, where AT is the transpose of A.
(b) Let f(x) = x · Ax, where A is an n × n matrix and x ∈ Rn. Show
that ∇f(x) = (A+AT )x.
(6) Find the closest point on the parabola y = (x+ 1)2 to the point a = (5, 2).
Hint: start by parametrizing the parabola as {(t, (t+1)2)|t ∈ R}. You want
to minimize the function f(t) = (t− 5)2 + ((t+ 1)2 − 2)2 which measures
the distance (squared) from points on the parabola to the point a. Use
the first order necessary conditions to find a candidate for the minimizer.
Does the point you found satisfy the second order necessary conditions for
a local min? What about the second order sufficient conditions? In this
question its ok if you only show the candidate is a local minimizer without
proving it is in fact the global minimizer.
Recall that a function of the form f(x) = 12x · Qx − b · x, where Q is a
positive definite n × n symmetric matrix and x ∈ Rn, can be written in
the form f(x) = 12 (x−x∗) ·Q(x−x∗)− 12x∗ ·Qx∗, where x∗ = Q−1b. This
is what is called ‘completing the square’. The next question explores a
function which, for some α, has this form.
(7) Given α ∈ R, let fα : R2 → R be given as fα(x, y) = (α − 2)x2 + 4αy2 +
3xy − 5y.
(a) For each α find the point(s), if any, satisfying the first order necessary
condition for a local minimum of fα.
(b) For which α do the points you found in (a) satisfy the second order
necessary condition for a local minimum of fα?
(c) For which α do the points you found in (a) satisfy the second order
sufficient condition for a local minimum of fα?
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(d) For which α do the points you found in (c) are in fact a global
minimum. Prove your claim.
Hint: it may be useful to complete the square.
(8) Find the local minimum point(s) for the function f(x, y) = (y2 − x)2 on
the set {(x, y) ∈ R2|y ≥ x+ 2} by finding the point(s) which satisfy the
necessary first and second order conditions for a local minimum. Prove
that your solution is actually a global minimum.
Here is one approach you can try: after you find the candidate for local
min, study how the function varies in a feasible direction (u, v).
The following six problems—the ones indexed by Roman numerals—are for
practice only and are not to be turned in.
I. [J] Chapter 1.2: Less basic problems: 2.
II. [J] Chapter 1.2: Less basic problems: 8. Please prove without using
Theorem 5 but by giving an ε− δ argument.
III. [J] Chapter 1.4: Less basic problems: 1.
IV. Show that any n× n matrix of the form xxT , where xT = (x1, ..., xn) is a
row vector, is positive semidefinite.
V. Let Hk := {(x, y) ∈ R2|f(x, y) = k)}denote the k-level sets of the function
f(x, y) = xy.
(a) Draw the level sets H1, H0, H1.
(b) Draw the level sets of f near the point (1, 0). Do they look like layers
of an onion? What about the level sets of f near the point (0, 0)?
(c) Can the level set H1 be expresses as the graph of a function of x near
the point (1, 1)? What about as a function of y? If it can, do so,
express it as a function of x and/or y. If not explain why.
(d) Can the level set H0 be expresses as the graph of a function of x near
the point (0, 0)? What about as a function of y? If it can, do so,
express it as a function of x and/or y. If not, explain why.
VI. Let f : R2n → R be defined as f(x, y) = 1/2|Ax − By|2, where A and B
are m× n matrices, x, y ∈ Rn. Find ∇f(x, y) and ∇2f(x, y).