EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 7 solutions
9.25 Smoothed t to given data. Consider the problem
minimize f (x) =
n
i=1
(xi yi ) +
n1
i=1 (xi+1
xi ) 2
where > 0 is smoothing parameter, is a convex penalty function, and x Rn is
the va

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 5 solutions
6.9 Minimax rational function tting. Show that the following problem is quasiconvex:
minimize
max
i=1,.,k
p(ti )
yi
q(ti )
where
p(t) = a0 + a1 t + a2 t2 + + am tm ,
q(t) = 1 + b1 t + + bn

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 8 solutions
8.16 Maximum volume rectangle inside a polyhedron. Formulate the following problem as a
convex optimization problem. Find the rectangle
R = cfw_x Rn | l
x
u
of maximum volume, enclosed in a

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 6 solutions
7.8 Estimation using sign measurements. We consider the measurement setup
yi = sign(aT x + bi + vi ),
i
i = 1, . . . , m,
where x Rn is the vector to be estimated, and yi cfw_1, 1 are the me

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 4 solutions
5.5 Dual of general LP. Find the dual function of the LP
minimize cT x
subject to Gx h
Ax = b.
Give the dual problem, and make the implicit equality constraints explicit.
Solution.
(a) The L

EE364a, Winter 2007-08
Prof. S. Boyd
EE364a Homework 6 solutions
6.9 Minimax rational function tting. Show that the following problem is quasiconvex: minimize where p(t) = a0 + a1 t + a2 t2 + + am tm , q(t) = 1 + b1 t + + bn tn , max p(ti ) yi q(ti )
i=1,

Additional Exercises for Convex Optimization
Stephen Boyd
Lieven Vandenberghe
January 10, 2014
This is a collection of additional exercises, meant to supplement those found in the book Convex
Optimization, by Stephen Boyd and Lieven Vandenberghe. These ex

Convex Optimization Boyd & Vandenberghe
2. Convex sets
ane and convex sets
some important examples
operations that preserve convexity
generalized inequalities
separating and supporting hyperplanes
dual cones and generalized inequalities
21
Ane set
l

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 3 solutions
3.36 Derive the conjugates of the following functions.
(a) Max function. f (x) = maxi=1,.,n xi on Rn .
Solution. We will show that
f (y) =
0 if y 0, 1T y = 1
otherwise.
We rst verify the do

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 1 solutions
2.5 What is the distance between two parallel hyperplanes cfw_x Rn | aT x = b1 and
cfw_x Rn | aT x = b2 ?
Solution. The distance between the two hyperplanes is |b1 b2 |/ a 2 . To see this,

EE364a, Winter 2013-14
Prof. S. Boyd
EE364a Homework 2 solutions
3.2 Level sets of convex, concave, quasiconvex, and quasiconcave functions. Some level sets
of a function f are shown below. The curve labeled 1 shows cfw_x | f (x) = 1, etc.
3
2
1
Could f b