bv_cvxbook_extra_exercises - Additional Exercises for...

Additional Exercises for Convex Optimization
Stephen Boyd Lieven Vandenberghe
June 15, 2013
This is a collection of additional exercises, meant to supplement those found in the book Convex
Optimization , by Stephen Boyd and Lieven Vandenberghe. These exercises were used in several
courses on convex optimization, EE364a (Stanford), EE236b (UCLA), or 6.975 (MIT), usually for
homework, but sometimes as exam questions. Some of the exercises were originally written for the
book, but were removed at some point. Many of them include a computational component using
CVX, available at www.stanford.edu/~boyd/cvx/ . Matlab files required for these exercises can
be found at the book web site www.stanford.edu/~boyd/cvxbook/ . Some of the exercises require
a knowledge of elementary analysis.
You are free to use these exercises any way you like (for example in a course you teach), provided
you acknowledge the source. In turn, we gratefully acknowledge the teaching assistants (and in
some cases, students) who have helped us develop and debug these exercises. Pablo Parrilo helped
develop some of the exercises that were originally used in 6.975.
Course instructors can obtain solutions by request to [email protected] , or by email
to us. In either case please specify the course you are teaching and give its URL.
We’ll update this document as new exercises become available, so the exercise numbers and
sections will occasionally change. We have categorized the exercises into sections that follow the
book chapters, as well as various additional application areas. Some exercises fit into more than
one section, or don’t fit well into any section, so we have just arbitrarily assigned these.
Stephen Boyd and Lieven Vandenberghe
1

Contents
1 Convex sets 3
2 Convex functions 5
3 Convex optimization problems 13
4 Duality 26
5 Approximation and fitting 39
6 Statistical estimation 52
7 Geometry 59
8 Unconstrained and equality constrained minimization 74
9 Interior point methods 80
10 Mathematical background 85
11 Circuit design 86
12 Signal processing and communications 93
13 Finance 103
14 Mechanical and aerospace engineering 116
15 Graphs and networks 127
16 Energy and power 135
17 Miscellaneous applications 146
2

1 Convex sets
1.1 Is the set { a ∈ R
k
| p (0) = 1 , | p ( t ) | ≤ 1 for α ≤ t ≤ β } , where
p ( t ) = a
1
+ a
2
t + · · · + a
k
t
k − 1
,
convex?
1.2 Set distributive characterization of convexity. [7, p21], [6, Theorem 3.2] Show that C ⊆ R
n
is
convex if and only if ( α + β ) C = αC + βC for all nonnegative α , β .
1.3 Composition of linear-fractional functions . Suppose φ : R
n
→ R
m
and ψ : R
m
→ R
p
are the
linear-fractional functions
φ ( x ) =
Ax + b
c T x + d
, ψ ( y ) =
Ey + f
g T y + h
,
with domains dom φ = { x | c
T
x + d> 0 } , dom ψ = { y | g
T
x + h> 0 } . We associate with φ and
ψ the matrices
bracketleftBigg
A b
c
T
d
bracketrightBigg
,
bracketleftBigg
E f
g
T
h
bracketrightBigg
,
respectively.