Linear Programming Computational Geometry, WS 2007/08 Lecture 7 Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät

Linear Programming

Computational Geometry, WS 2007/08Lecture 7

Prof. Dr. Thomas Ottmann

Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg

Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 2

Overview

• Problem formulation and example.• Incremental, deterministic algorithm.• Randomized algorithm.• Unbounded linear programs.• Linear programming in higher dimensions.


Problem Description

Maximize c1x1 + c2x2 + ... + cdxd

Subject to the conditions: a1,1x1 + ... a1,dxd b1

a2,1x1 + ... a2,dxd b2

: : :an,1x1 + ... an,dxd bn

Linear program of dimension d: c = (c1,c2,...,cd)hi = {(x1,...,xd) ; ai,1x1 + ... + ai,dxd bi}

li = hyperplane that bounds hi ( straight lines, if d=2 )H = {h1, ... , hn}


Example

• Production of two goods A and B using four raw materials • Value of A: 6 CU, value of B: 3 CU

Rm1 Rm2 Rm3 Rm4

Prod A

Prod B

Reserve

2

4

5

2 1

2

6 2

4

2 2

3

Maximize profit: fc (x) = 6xA + 3xB under the conditions:

xA = 0, xB xA, xB = 02xA + 4xB 52xA + 1xB 26xA + 2xB 42xA + 2xB 3 xA, xB 0


Chart

Ii (xA, xB) (xA, xB)

1 (0, 5/4) (5/2, 0)

2 (0, 2) (1, 0)

3 (0, 2) (2/3, 0)

4 (0, 3/2) (3/2, 0)

2

3/2

5/41

1/2

1/22/3 1 3/2 2 5/2xB

xA


Chart


Structure of Feasible Region

1. Bounded

2. Unbounded

3. Empty

C

C

C


Result

Four possibilities for the solution of a linear program

1. A vertex of the feasible region is the only solution. 2. One edge of the feasible region contains all

solutions. 3. There are no solutions. 4. The feasible region is unbounded toward the

direction of optimization.

In case 2: Choose the lexicographically minimum solution corner


Bounded Linear Programs

Assumption :

Algorithm UnboundedLP(H, c) yields either

a) a ray in H, which is unbounded towards c, or

b) two half planes h1 and h2, so that h1 h2 is bounded towards c, or

c) the answer, that LP(H, c) has no solution, because the feasible region is empty.


Incremental Algorithm

Let C2 = h1 h2

Remaining half planes: h3,..., hn

Ci = Ci-1 hi = h1 ... hi

Compute-optimal-vertex (H, c)

v2 := l1 l2 ; C2 := h1 h2

for i := 3 to n do

Ci := Ci-1 hi

vi := optimal vertex of Ci

C2 C3 C4 ... Cn = C

Ci = C =


Optimal Vertex

Lemma 1: Let 2 < i n, then we have :

1.If vi-1 hi, then vi = vi-1.

2.If vi-1 hi, then either Ci = or vi li, where li is the line bounding hi.


Optimal Vertex

h1

h2

h3

h4

v3= v4

c

h1

h2

h3

h4

h5

c

v4 v5


Next Optimal Vertex

fc(x) = c1x1 + c2x2

vi-1

Ci-1


Algorithm: 2D-LPInput: A 2-dimensional Linear Program (H, c )Output: Either one optimal vertex or or a ray

along which (H, c ) is unbounded.

if UnboundedLP(H, c ) reports (H, c ) is unbounded or infeasible then return UnboundedLP(H, c )

else report h1 := h; h2 := h´ ; v2 := l1 l2 let h3,...,hn be the remaining half-planes of Hfor i:= 3 to n do if vi-1 hi then vi := vi-1

else Si-1 := Hi-1 * livi := 1-dim-LP(Si-1, c )

if vi not exists then return return vn

Running time: O(n²)


Algorithm 1D-LP


Algorithm: 1D-LP

Find the point x on li that maximizes fc(x) , subject to

the constraints x hj, for 1 j i –1

Observation: li hj is a ray

Let si-1 := { h1 li, ..., hi-1 li}

Algorithm 1D-LP{si-1, c }

p1 = s1

for j := 2 to i-1 do pj = pj-1 sj

if pi-1 then return the optimal vertex of pi-1 else return

Time: O(i)


Addition of Halfplanes in Different Orders

1 23 4

5 67 89 10

11 12

vi

Good

1 23 4

5 67 8

9 1011 12

Bad


Optimal Vertex

h1

h2

h3

h4

v3= v4

c

h1

h2

h3

h4

h5

c

v4 v5


Randomized Algorithm: 2D-LPInput: A 2-dimensional Linear Program (H, c )Output: Either one optimal vertex or or a ray

along which (H, c ) is unbounded.

if UnboundedLP(H, c ) reports (H, c ) is unbounded or infeasible then return UnboundedLP(H, c )

else report h1 := h; h2 := h´ ; v2 := l1 l2 let h3,...,hn be the remaining half-planes of H

compute a random permutation h3, ..., hn

for i:= 3 to n do if vi-1 hi then vi := vi-1

else Si-1 := Hi-1 * livi := 1-dim-LP(Si-1, c )

if vi not exists then return return vn


Randomization

Theorem:

The 2-dimensional linear programming problem with nconstraints can be solved in O(n) randomizedexpected time using worst-case linear storage.


Random Variable Xi

n

i

XiO3

i)(

n

i

i

n

i

i XEiXiE33

)()(

E[xi] is the probability that vi-1 hi

Xi = 1, if vi-1 hi, and Xi = 0, otherwise


Unbounded Linear Programs

imin, an index with i min = min j,1 j n

i := The smaller angle that i makes with c

i : The outward normal of hi

hi

i

i

c


Unbounded Linear Programs

Hmin := { hj H | j = i min}

Hpar := { hj H | j = - i min}

Hpar {

Hmin {

hi*c


Lemma

Let H = {h1,h2,...,hn} be a set of half-planes.

Assuming that (Hmin Hpar) is not empty.

1. If li* hj is unbounded in the direction c for every half-plane hj

in the set H \ (Hmin Hpar), then (H, c ) is unbounded along a ray contained in li* .

2. If li* hj* is bounded in the direction c for some hj* in

H \ (Hmin Hpar), then the linear program ({hi* , hj*}, c )

is bounded.


Algorithm: Unbounded-LP

Input: A 2-Dimensional Linear Program (H, c )

Output: Either a pair of halfplanes bounding the LP or or a ray along which (H, c) is unbounded.

1. For each half plane hi H compute j 2. Let hi be half plane with i = min j,1 j n3. Hmin := { hj H | j = min}4. Hpar := { hj H | j = - min}5. Ĥ = H \ (Hmin Hpar), compute intersection in Hmin Hpar

6. If the intersection is empty then report (H, c ) is infeasible

else Let hi Hmin be the half-plane whose line bounds the intersection

if there is half plane hj* Ĥ such that li*hj* bounded in cthen report ({hi* , hj*}, c ) is boundedelse report is bounded along li* ( Ĥ )


Higher Dimensions

Let h1,...,hd H be the d certificate half-spaces that UnboundedLP returns.

Ci := h1 h2 ... hi

Lemma: Let d < i n, and let Ci be defined as above.

1. If vi-1 hi, then vi = vi-1

2. If vi-1 hi, then either Ci= or vi gi, where gi is the

hyperplane that bounds hi.


Algorithm: Randomized-LPInput : A linear program (H, c ).

Output : Either one optimal vertex or or a ray along which (H, c ) is unbounded.

if UnboundedLP(H, c ) reports (H, c ) is unbounded

then Report a ray along which (H, c ) unbounded.

else Let h1,...,hd H be the certificate halfplanes returned by UnboundedLP, and let vd be their vertex of intersection

Compute a random permutation hd+1,...,hn

for i = d+1 to n do if vi-1 hi

then vi = vi-1

else Si-1:= Hi-1 * bd(hi)

vi:= d-1-dim-LP(Si-1, c ) if vi not exists then return

return vn


Theorem

The d-dimensional linear programming problem with

n constraints can be solved in O(d!n) expected time

using linear storage.

Documents

Linear Programming Computational Geometry, WS 2007/08 Lecture 7 Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät