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
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
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
Optimiza
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 objecti
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 optimiz
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. Ther
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 concav
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
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 + vi
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) i
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
Convex Optimization Boyd & Vandenberghe
2. Convex sets
ane and convex sets
some important examples
operations that preserve convexity
generalized inequalities
separating and supporting hyperplane
Convex Optimization Boyd & Vandenberghe
3. Convex functions
basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
log-concave and log-conve
Convex Optimization Boyd & Vandenberghe
12. Interior-point methods
inequality constrained minimization
logarithmic barrier function and central path
barrier method
feasibility and phase I methods
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 a
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
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
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 s
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 (
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