EE236A (Fall 2013-14)
Lecture 9
Structural optimization
minimum weight truss design
topology design
limit analysis
91
Truss
m bars (members), N nodes (joints)
length of bar i is li, cross-sectional area xi
nodes n + 1, . . . , N are anchored
extern
EE236A (Fall 2013-14)
Lecture 10
FIR lter design
linear phase lter design
magnitude lter design
equalizer design
101
Finite impulse response (FIR) lter
n1
y(t) =
=0
h u(t )
u : Z R is input signal; y : Z R is output signal
hi R are lter coecients; n
EE236A (Fall 2013-14)
Lecture 1
Introduction
course overview
linear optimization
examples
history
approximate syllabus
basic denitions
linear optimization in vector and matrix notation
halfspaces and polyhedra
geometrical interpretation
11
Linear opt
EE236A (Fall 2013-14)
Lecture 6
Duality
dual of an LP in inequality form
variants and examples
complementary slackness
61
Dual of linear program in inequality form
we dene two LPs with the same parameters c Rn, A Rmn, b Rm
an LP in inequality form
min
EE236A (Fall 2013-14)
Lecture 4
Convexity
convex hull
polyhedral cone
decomposition
41
Convex combination
a convex combination of points v1, . . . , vk is a linear combination
x = 1 v 1 + 2 v 2 + + k v k
with i 0 and
k
i=1 i
=1
for k = 2, the point x i
EE236A (Fall 2013-14)
Lecture 11
Control applications
optimal input design
pole placement with low-authority control
111
System model
y(t) = h0u(t) + h1u(t 1) + h2u(t 2) +
u(t) is input, y(t) is output, (h0, h1, . . .) is impulse response
matrix descri
EE236A (Fall 2013-14)
Lecture 8
Linear-fractional optimization
linear-fractional program
generalized linear-fractional program
examples
81
Linear-fractional program
cT x + d
minimize
gT x + h
subject to Ax b
gT x + h 0
if needed, we interpret a/0 as a
EE236A (Fall 2013-14)
Lecture 2
Piecewise-linear optimization
piecewise-linear minimization
1- and -norm approximation
examples
modeling software
21
Linear and ane functions
linear function: a function f : Rn R is linear if
f (x + y) = f (x) + f (y)
x
EE236A (Fall 2013-14)
Lecture 12
Simplex method
adjacent extreme points
one simplex iteration
cycling
initialization
implementation
121
Problem format and assumptions
minimize cT x
subject to Ax b
A has size m n
assumption: the feasible set is nonemp
EE236A (Fall 2013-14)
Lecture 3
Polyhedra
linear algebra review
minimal faces and extreme points
31
Subspace
denition: a nonempty subset S of Rn is a subspace if
x, y S,
, R
=
x + y S
extends recursively to linear combinations of more than two vectors:
EE236A (Fall 2013-14)
Lecture 5
Alternatives
theorem of alternatives for linear inequalities
Farkas lemma and other variants
51
Theorem of alternatives for linear inequalities
for given A, b, exactly one of the following two statements is true
1. there