Truss
EE236A (Fall 2007-08)
Lecture 5
Structural optimization
m bars with lengths li and cross-sectional areas xi
N nodes; nodes 1, . . . , n are free, nodes n + 1, . . . , N are anchored
external load: forces fi R2 at nodes i = 1, . . . , n
minimum w
L. Vandenberghe
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) =
L. Vandenberghe
EE236A Fall 2013
Homework assignment #8
In this problem you are asked to write a MATLAB function for solving 1 -norm approximation problems
minimize P u + q 1
(1)
via the primal-dual interior-point method of lecture 15. The function will b
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 16
Self-dual formulations
self-dual linear programs
self-dual embedding
interior-point method for self-dual embedding
161
Optimality and infeasibility
maximize bT z
subject to AT z + c = 0
z0
minimize cT x
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 15
Primal-dual interior-point method
primal-dual central path equations
infeasible primal-dual method
151
Optimality conditions
primal and dual problem
maximize bT z
subject to AT z + c = 0
z0
minimize cT x
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 18
Integer linear programming
a few basic facts
branch-and-bound
181
Denitions
integer linear program (ILP)
minimize cT x
subject to Ax b
x Zn
c
mixed integer linear program: only some of the variables are i
L. Vandenberghe
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 interpretati
60 Chapter 3. The Simplex Method
WW
Exercise 3-3-6. Solve the following linear program. is the fcasibie region unbounded? Is
the solution set unbounded? ifso, nd vectors a and a such that it + Au is optimal for all
120.
min 2: = 3.7m 3x2 2x3 + 5x4
subject
Fall 2015
EE 236A
Prof. Christina Fragouli
Review Problems from Last Years
Midterm and Final
1
Problem Assume in the following problem that the constraint Ax b implies that g T x + h > 0.
minimize
cT x+d
g T x+h
subject to Ax b
g T x + h 0,
(1)
where A Rm
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 14
Barrier method
centering problem
Newton decrement
local convergence of Newton method
short-step barrier method
global convergence of Newton method
predictor-corrector method
141
Centering problem
cent
L. Vandenberghe
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 feasib
L. Vandenberghe
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 t
Linear Programming Exercises
Lieven Vandenberghe
Electrical Engineering Department
University of California, Los Angeles
Fall Quarter 2013-2014
1
Hyperplanes and halfspaces
Exercise 1. When does one halfspace contain another? Give conditions under which
c
L. Vandenberghe
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 i
L. Vandenberghe
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 =
L. Vandenberghe
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 ine
L. Vandenberghe
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
L. Vandenberghe
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 in
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 7
Duality II
sensitivity analysis
two-person zero-sum games
circuit interpretation
71
Sensitivity analysis
purpose: extract from the solution of an LP information about the
sensitivity of the solution with
L. Vandenberghe
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 l
L. Vandenberghe
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 respon
L. Vandenberghe
EE236A (Fall 2013-14)
Lecture 13
The central path
nonlinear optimization methods for linear optimization
logarithmic barrier
central path
131
Ellipsoid method
ellipsoid algorithm
a general method for (nonlinear) convex optimization, in
Fall 2015
EE 236A
Christina Fragouli
Linear Programming
Shortest Path
Problem: A weighted directed graph, which contains 8 vertices and 15 edges, is shown in the figure
below. There is a single source s. The weight of an edge from vertex u to v can be see
Fall 2015
EE 236A
Christina Fragouli
Linear Programming
Homework 7
Due 8:10am, Tuesday Dec. 1, 2015
Problem 1 (4 points):
(a) Use the simplex procedure to solve the following problem
minimize z = x y
subject to x + y 2
x y 6
x, y 0.
(b) Draw a graphical r
Vectors
EE236A (Fall 2007-08)
Lecture 2
Linear inequalities
(column) vector x Rn:
x1
x
x = .2
.
xn
vectors
xi R: ith component or element of x
inner products and norms
also written as x = (x1, x2, . . . , xn)
linear equalities and hyperplanes
some
Linear program (LP)
EE236A (Fall 2007-08)
Lecture 1
Introduction and overview
n
minimize
cj xj
j =1
n
linear programming
aij xj bi,
i = 1, . . . , m
cij xj = di,
subject to
i = 1, . . . , p
j =1
n
example from optimal control
j =1
example from combinat
EE236A (Winter 2012-13)
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 o