EE364: Convex Optimization
Section 3
April 25, 2005
EE364 3
Outline
Generalized eigenvalues
Hyperbolic constraints
Homework hints
Conjugate function example
Proof of Hlders inequality
o
EE364 3
1
Generalized eigenvalues
the maximum generalized eigenv
New York University
Department of Economics
V31.0006
Mathematics for Economists
C. Wilson
September 15, 2011
Concave and Quasi-Concave Functions
A set X Rn is convex if x, y X implies x + (1 ) y X for all [0, 1] .
Geometrically, if x, y Rn , then cfw_z Rn
New York University
Department of Economics
V31.1006
Mathematics for Economists
C. Wilson
March 8, 2011
Sets and Functions
x = y means that the value of x is the same as the value of y .
x y means that the denition of x is the same as the denition of y .
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 2
1. In the rst lecture, in the problem of designing lamp powers to illuminate a set of surfaces,
we encountered the function
f (p) = max | log aT p log Ides |,
i
i
where ai Rm and Ides > 0 are given,
40
CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
Figure 1.16: Line through P0 parallel to !
v
1.5
Equations of Lines and Planes in 3-D
Recall that given a point P = (a; b; c), one can draw a vector from the origin
to P . Such a vector is called the positio
Chapter 3
Basic Properties of Convex Sets
3.1
Convex Sets
Convex sets play a very important role in geometry. In
this chapter, we state some of the classics of convex
ane geometry: Carathodorys Theorem, Radons Thee
orem, and Hellys Theorem.
These theorems
CS599: Convex and Combinatorial Optimization
Fall 2013
Lectures 5-6: Convex Functions
Instructor: Shaddin Dughmi
Announcements
HW1 is out, due Thursday 9/26
Make sure you get email from me
Today: Convex Functions
Read all of B&V Chapter 3.
Outline
1
Conve
Linear Algebra & Analysis Review
UW EE/AA/ME 578 Convex Optimization
January 9, 2015
1
Notation
1. Book pages without explicit citation refer to [1].
2. Rn denotes the set of real n-column vectors.
3. Rmn denotes the set of real m n real matrices.
4. Sn d
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 1
1. Suppose C Rn (not necessarily convex) and 0 C. Dene C to be:
C = cfw_u | u, x 1 for all x C,
and let C = (C ) . Here u, x = uT x.
(a) Prove that C is a closed convex set that contains the origin.
APPENDIX A:
Notation and Mathematical
Conventions
In this appendix we collect our notation, and some related mathematical
facts and conventions.
A.1
SET NOTATION AND CONVENTIONS
If X is a set and x is an element of X, we write x 2 X. A set can be
specied
CONVEXITY AND OPTIMIZATION
1. CONVEX
SETS
1.1. Denition of a convex set. A set S in Rn is said to be convex if for each x1 , x2 S, the line segment x1 + (1-)x2 for (0,1) belongs to S. This says that all points on a line connecting two points in the set ar
EE364 Convex Optimization June 7 8 or June 8 9, 2006.
Prof. S. Boyd
Final exam solutions
1. Optimizing processor speed. A set of n tasks is to be completed by n processors. The variables to be chosen are the processor speeds s1 , . . . , sn , which must l
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 3
1. Show that the following functions f : Rn R are convex.
(a) The dierence between the maximum and minimum value of a polynomial on a given
interval, as a function of its coecients:
f (x) = sup p(t)
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 4
1. Suppose that the random variable X on Rn has log-concave density, and let Y = g(X),
where g : Rn R. For each of the following statements, either give a counter-example, or
show that the statement
EE 578, Univ of Washington, Winter 2013
7. Duality
Lagrange dual problem
weak and strong duality
geometric interpretation
optimality conditions
perturbation and sensitivity analysis
examples
generalized inequalities
71
Lagrangian
standard form prob
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 6
1. Note that x Rn , y, z R satisfy
xT x yz, y 0, z 0
if and only if
2x
yz
y + z, y 0, z 0.
2
Use this observation to cast the following problems as SOCPs.
(a) Maximizing harmonic mean.
m
maximize
i
EE 578, Univ of Washington, Winter 2015
5. Linear and quadratic programming
linear programming
quadratic programming
second-order cone programming
51
Linear program (LP)
minimize cT x + d
subject to Gx h
Ax = b
convex problem with ane objective and co
EE363
Prof. S. Boyd
Solving semidenite programs using cvx
There are now many software packages that solve SDPs eciently, once youve put the
problem into a standard format. But writing and debugging code that transforms your
problem to a standard format ca
EE364a, Winter 2007-08
Prof. S. Boyd
EE364a Homework 5 solutions
4.15 Relaxation of Boolean LP. In a Boolean linear program, the variable x is constrained
to have components equal to zero or one:
minimize cT x
subject to Ax b
xi cfw_0, 1,
(1)
i = 1, . . .
EE364a, Winter 2007-08
Prof. S. Boyd
EE364a Homework 2 solutions
2.28 Positive semidenite cone for n = 1, 2, 3. Give an explicit description of the positive
semidenite cone Sn , in terms of the matrix coecients and ordinary inequalities, for
+
n = 1, 2, 3
EE364a, Winter 2007-08
Prof. S. Boyd
EE364a Homework 7 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 2007-08
Prof. S. Boyd
EE364a Homework 4 solutions
4.11 Problems involving 1 - and -norms. Formulate the following problems as LPs. Explain in detail the relation between the optimal solution of each problem and the
solution of its equivalen
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,
EE/AA/ME 578
Univ. of Washington, Winter 2015
Homework 5
1. Consider the following problem.
(
)
sup (1/2)xT P x + q T x + r
minimize
subject to
P
Ax b.
where is the set of possible matrices P . Assume that P is uncertain, and the other
parameters (q, r,
EE364a Convex Optimization I
March 1415 or March 1516, 2008.
Prof. S. Boyd
Final exam solutions
You may use any books, notes, or computer programs (e.g., Matlab, cvx), but you may not
discuss the exam with anyone until March 18, after everyone has taken t