L. Vandenberghe
EE236B Winter 2015
Additional problem for homework #5
Generalized posynomials. In lecture 4 we dened a posynomial as a function of the form
m
f (x1 , . . . , xn ) =
a
xj ij ,
i
i=1
dom f = Rn ,
+
j=1,.,n
with i > 0. The exponents aij can t
Additional Exercises for Convex Optimization
Stephen Boyd
Lieven Vandenberghe
January 17, 2017
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
L. Vandenberghe
EE236B Winter 2015
Homework assignment #1
This assignment is an introduction to the MATLAB software package CVX that will be
used in the course. CVX can be downloaded from www.cvxr.com.
Python users are welcome to use CVXPY (cvxpy.org) ins
L. Vandenberghe
February 25, 2016
EE236B homework #6 solutions
1. Exercise T5.26.
(a) The figure shows the feasible set (the intersection of the two shaded disks) and
some contour lines of the objective function. There is only one feasible point,
(1, 0),
L. Vandenberghe
February 11, 2016
EE236B homework #4 solutions
1. Exercise A3.5. We describe two solutions for this problem.
The first solution is to show that the problem is equivalent to the optimization problem
max (aTi y + bi t)
minimize
i=1,.,m
subje
L. Vandenberghe
February 18, 2016
EE236B homework #5 solutions
1. Exercise T4.43 (b, c).
(b) The inequality 1 (x) t1 holds if and only if A(x) t1 I, and m (A(x) t2
holds if and only if A(x) t2 I. Therefore we can minimize 1 m by solving
minimize t1 t2
sub
L. Vandenberghe
February 4, 2016
EE236B homework #3 solutions
1. Exercise T3.2. The first function could be quasiconvex because the sublevel sets that
are shown are convex. It is definitely not concave or quasiconcave because the superlevel sets are not c
L. Vandenberghe
January 21, 2016
EE236B homework #2 solutions
1. Exercise T2.12 (d,e,g).
(d) For fixed y, the set cfw_x | kx x0 k2 kx yk2 is a halfspace. This can be seen
by squaring the two sides of the inequality kx x0 k2 kx yk2 :
xT x 2xT0 x + xT0 x0
L. Vandenberghe
January 14, 2016
EE236B homework #1 solutions
1. (a) Expanding the squares in the ith term of the cost function gives
(ui uc )2 + (vi vc )2 R2 = 2ui uc 2vi vc + u2c + vc2 R2 + u2i + vi2
= 2ui uc 2vi vc + w + u2i + vi2 .
This is linear in u
L. Vandenberghe
March 10, 2016
EE236B homework #8 solutions
1. Exercise A8.9.
(a) The problem is
minimize
m
X
yi =1
log (bi + aTi x)
m
X
yi =1
log (bi aTi x).
where
1 Z u t2 /2
e
dt.
(u) =
2
(u) is log-concave (it is the cumulative distribution functi
L. Vandenberghe
March 15, 2011
EE236B Final Exam
Problem 1 (20 points). Show that the following functions f : Rn R are convex.
1. f (x) = exp(g(x) where g : Rn R has a convex domain and satisfies
"
2 g(x) g(x)
g(x)T
1
#
0
for x dom g.
2. The function
f (x
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
Convex Optimization Boyd & Vandenberghe
1. Introduction
mathematical optimization
least-squares and linear programming
convex optimization
example
course goals and topics
nonlinear optimization
brief history of convex optimization
11
Mathematical o
Convex Optimization Boyd & Vandenberghe
6. Approximation and tting
norm approximation
least-norm problems
regularized approximation
robust approximation
61
Norm approximation
minimize
(A Rmn with m n,
Ax b
is a norm on Rm)
interpretations of solution
Convex Optimization Boyd & Vandenberghe
3. Convex functions
basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
log-concave and log-convex functions
convexity with respect to generalized ineq
Convex Optimization Boyd & Vandenberghe
8. Geometric problems
extremal volume ellipsoids
centering
classication
placement and facility location
81
Minimum volume ellipsoid around a set
Lwner-John ellipsoid of a set C: minimum volume ellipsoid E s.t. C
Convex Optimization Boyd & Vandenberghe
7. Statistical estimation
maximum likelihood estimation
optimal detector design
experiment design
71
Parametric distribution estimation
distribution estimation problem: estimate probability density p(y) of a
ran
Convex Optimization Boyd & Vandenberghe
12. Interior-point methods
inequality constrained minimization
logarithmic barrier function and central path
barrier method
feasibility and phase I methods
complexity analysis via self-concordance
generalized
Convex Optimization Boyd & Vandenberghe
13. Conclusions
main ideas of the course
importance of modeling in optimization
131
Modeling
mathematical optimization
problems in engineering design, data analysis and statistics, economics,
management, . . . ,
L. Vandenberghe
February 16, 2012
EE236B homework #5 solutions
1. Exercise T4.27. To show the equivalence with the problem in the hint, assume x 0 is
fixed in the hint and optimize over v and w. This is a quadratic problem with equality
constraints. The o
L. Vandenberghe
February 9, 2012
EE236B homework #4 solutions
1. Exercise T4.13. We consider the constraints separately. The ith constraint is
n
X
Aij xj bi
for all Aij [Aij Vij , Aij Aij + Vij ].
j=1
Given x, the expression on the left-hand side of the i
L. Vandenberghe
January 26, 2017
EE236B homework #2 solutions
1. Exercise T2.37 (b,c).
(b) A vector x R2k+1 and a symmetric matrix Y Sk+1 satisfy
xi =
X
Ymn ,
i = 1, . . . , 2k + 1,
(1)
m+n=i+1
(the sum on the right-hand side is over all m and n with m +
L. Vandenberghe
February 19, 2015
EE236B homework #6 solutions
1. Exercise A3.5. We show that the problem is equivalent to the optimization problem
max (aTi y + bi t)
minimize
i=1,.,m
subject to
min (cTi y + di t) 1
i=1,.,p
(1)
F y gt
t0
with variables y,
HW1 Wenyuan Li UID:304380108 [email protected]
Homework assignment #1
We consider the illumination problem of lecture 1. We take
p max =1
I des 1
and
, so the problem is
with variable
where
pR
m
min imize
f 0 ( p) max log(akT p)
subject to
0 p j 1,
L. Vandenberghe
January 13, 2011
EE236B homework #1 solutions
1. Exercise A10.1
(a) First note that the inequality is trivially satisfied if b = 0. Assume b 6= 0. We
write g as
g(t) = (a tb)T (a tb) = kak22 + 2taT b + t2 kbk22 .
Setting the derivative equ