ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 7
Date: 11/05/2012
Convex Optimization
Spring 2012
Exercise 5.5 The Lagrangian is
L(x, , ) = cT x + T (Gx h) + T (Ax b)
= (cT + T G + T A)x hT T b
which is a
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 6
Date: 04/05/2012
Convex Optimization
Spring 2012
Exercise 4.8 b)
This problem is always feasible. The vector c can be decomposed into a component parallel
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 5
Date: 20/04/2012
Convex Optimization
Spring 2012
Exercise 4.3
We verify that x satises the optimality condition (4.21). The gradient of the objective funct
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 4
Date: 23/03/2012
Convex Optimization
Spring 2012
Exercise 3.32
(a)We prove the result by verifying Jensens inequality. f and g are positive and convex, hen
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 1
Date: 02/03/2012
Convex Optimization
Spring 2012
Exercise 3.9
a)We know that a (twice dierentiable) function is convex if and only if dom f is convex and i
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 2
Date: 09/03/2012
Convex Optimization
Spring 2012
Exercise 3.3
Assume we are given any x1 , x2 domf and 0 1. First of all, due to convexity of f (x), we
hav
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions 1
Date: 02/03/2012
Convex Optimization
Spring 2012
Exercise 2.2
Convex sets:
The intersection of two convex sets is convex. Therefore if S is a convex set, t
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Solutions
Date: 18/05/2012
Convex Optimization
Spring 2012
Exercise 5.12
We derive the dual of the problem
m
minimize
log yi
i=1
(1)
subject to y = b Ax,
where A Rmn h
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 7
Date: 11/05/2012
Convex Optimization
Spring 2012
Exercise 5.5
Dual of a general LP Find the dual function of the LP
minimize cT x
subject to Gx
h
Ax = b
Giv
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 6
Date: 04/05/2012
Convex Optimization
Spring 2012
Exercise 4.8 b) Give an explicit solution.
Minimizing a linear function over a halfspace.
minimize cT x
sub
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 5
Date: 20/04/2012
Convex Optimization
Spring 2012
Exercise 4.3
Prove that x = (1, 1/2, 1) is optimal for the optimization problem
minimize (1/2)xT P x + q T
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 4
Date: 23/03/2012
Convex Optimization
Spring 2012
Exercise 3.32 Products and ratios of convex functions. In general the product or ratio of two
convex functi
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 3
Date: 16/03/2012
Convex Optimization
Spring 2012
Exercise 3.9 Second-order conditions for convexity on an ane set. Let F Rnm , x Rn . The
n R to the ane set
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problems 2
Date: 09/03/2012
Convex Optimization
Spring 2012
Exercise 3.3
Inverse of an increasing convex function. Suppose f : R R is increasing and convex on its
doma
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problem 1
Date: 02/03/2012
Convex Optimization
Spring 2012
Exercise 2.2
Show that a set is convex if and only if its intersection with any line is convex. Show that a
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
School of Computer and Communication Sciences
Problem Set
Date: 18/05/2012
Convex Optimization
Spring 2012
Exercise 5.12 Analytic centering. Derive a dual problem for
m
log(bi aT x)
i
minimize
i=1
cfw_x|aT x
i
wi
Applications for Convex Optimization
and Linear Programming
Spring Semester 2012
Lecturer:
Dr. Christina Fragouli, BC 126, e-mail: [email protected]
Special Topics Lecturer:
Dr. Nicholas Ruozzi
Teaching Assistants:
Marc Desgroseilliers, INR 140, e-