ADVANCED ALGORITHMS Lecture 20: Linear Programming 1 ANNOUNCEMENTS - - PowerPoint PPT Presentation

ADVANCED ALGORITHMS

Lecture 20: Linear Programming

ANNOUNCEMENTS

➤ HW 4 is due on Monday, November 5 ➤ Project meetings …

LAST CLASS

➤ Optimization ➤ Can phrase many natural problems as optimization — e.g. scheduling, matching,

shortest paths, …

➤ {x1, x2, …, xn} are variables — values in some domain D ➤ find maximum value of f(x) subject to


g2(x) ≥ 0 g1(x) ≥ 0 ….

WHEN CAN WE SOLVE OPTIMIZATION?

➤ The bad news: ➤ all the formulations we wrote so far are intractable!
 ➤ The good news: ➤ Continuous optimization with linear constraints, objective ➤ Convex optimization


Main challenge: can we express problem of interest as an optimization we can solve?

discrete optimization

domainis

a solitude

linear

programming

usually

Linear

Programs

In

Vars

x 1

2

Xu EIR

g

min

C H

t CzN

t

t Cnxn

subject

to

aTn 3 b

A

X t Akka t

tansen

b

n

gin

3 b

It

ain's.sn

Frivialobservations

air

b is

a hyperplane

a.tn

b

is

a half space

E

LAST CLASS (CONTD.)

➤ Linear and convex optimization ➤ Visualizing linear optimization

N t X tRy t 2 42 I

µ

Every linear constrain

is a half space

ATx b

A

X

t

a zXzt t 9nXn3

I

LINEAR FUNCTIONS — “LEVEL SETS”

x + 2y = 1 x + 2y = 1.5 x + 2y = 2

not

Yotz

Go Yo

f cT no yo

kot2y

y

Elnothyoth Is

I

2

Observation

minimizing

In

in

R

p

finding the furthest

R

point in R in the

direction

c

Ty

d

Finder

aTn Eb

ain

LEI

solve for the

n

OBSERVATIONS

➤ Theorem: optimum value is always achieved at a “vertex” or a “corner

point” of the polytope

➤ Feasible region (and hence opt value) can be unbounded ➤ Saw approach, not algorithm! ➤ Brute-force algorithm: check all corners — not all corners are feasible! ➤ Simplex algorithm (Dantzig 1949): systematic way to move from one

corner point to a neighbor — local search

EEE

77

ffeanhi.Y.IEaD

g

paenuaearnuiicheaEkh'bimff

7 Fia

melon

exp time

SIMPLEX

Algorithm

local search on set

• f corner points

start with

• ne

corner ft v

basic feasible solution

mom

that has a

random pivot

lowercturalmee

I

if there is

no such neighbor stop

return solution

am

x

U

l

v

In General

a

neighbor of

v

is

a ptithat has

precisely Cn l

equations in common with je

• ne of air

bi

repaced

Ain Eb

by another equation

ant'sbn

DOES “NO NEIGHBOR” => GLOBAL OPTIMUM?

i.e., can we get stuck?

➤ Note: ➤ this is NOT the same as local opt = global opt!


the

Fomenktopht

MATH OF LINEAR PROGRAMS — NEIGHBORING DIRECTIONS

we get

n different neighborsof

iix b

fit

• V

aIn bz

V

each

• btained by dropping

Hygagusataint ain bio replacing fight

ai'n bn

Z M

1.5

• nffittia

ai

Gen whatis AX

the i jthentry

ai

MATH OF LINEAR PROGRAMS — NEIGHBORING DIRECTIONS

fit

n'H

v

if I

a

ayy

• bi

bi

AX

is

a

diagonal matrix

Can think of

Ax

I

X

A

Moira

the directions UH

are simply the columns

• f

A

CORRECTNESS OF THE SIMPLEX ALGORITHM

• theorem

if

cTu c

ctu

ctfu.jo

v

then

3 j

s't

I

v

Iv

f

Iu

co

For any point atR claim

n

U

can be

expressed as

dj

U'd't when

dj 30

afs.bg

f

ai

f

g

um

ane

N

U

di Uli

because u

form a basis

for pi

ajtx

g.tv

di Cafu

g ajT

I I

11

bj

bj 1

because u

were

Y

chosentobe

columnsofthe

dj 70

inverse

GENERAL RESULTS

➤ [Dantzig 1949]: simplex algorithm — known to be exp-time ➤ Khachiyan’s “ellipsoid” algorithm — 1979

1

interior point

methods

Karmarkow 843

MATCHING AS LINEAR PROGRAM

MATCHING AS LINEAR PROGRAM

MATCHING AS LINEAR PROGRAM