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Linear Programming. Overview Formulation of the problem and example I ncremental, deterministic algorithm Randomized algorithm Unbounded linear programs Linear programming in higher dimensions.

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linear programming
Linear Programming
  • Overview
  • Formulation of the problem and example
  • Incremental, deterministic algorithm
  • Randomized algorithm
  • Unbounded linear programs
  • Linear programming in higher dimensions

Computational Geometry

Prof. Dr. Th. Ottmann

problem description

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

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

li = hyperplane that bounds hi (straight lines, ifd=2)

H = {h1, ... , hn}

Computational Geometry

Prof. Dr. Th. Ottmann

structure of the feasible region

C

C

C

Structure of the feasible region

1. Bounded

2. Unbounded

3. Empty

Computational Geometry

Prof. Dr. Th. Ottmann

bounded linear programs
Bounded linear programs
  • Assumption :
  • Algorithm UnboundedLP(H, c) yields either
  • a ray in∩H, which is unbounded towards c, or
  • two half planes h1 and h2, so that h1 h2 is bounded towards c, or
  • the answer, that LP(H, c) has no solution, because the feasible region is empty.

Computational Geometry

Prof. Dr. Th. Ottmann

incremental algorithm
Incremental algorithm

Let C2 = h1 h2

Remaininghalfplanes: 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 = 

Computational Geometry

Prof. Dr. Th. Ottmann

optimal vertex
Optimal Vertex

Lemma 1:Let2 < 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.

Computational Geometry

Prof. Dr. Th. Ottmann

optimal vertex7

h4

h1

h3

v3= v4

h2

h4

h1

h3

c

h5

c

h2

v4

v5

Optimal vertex

Computational Geometry

Prof. Dr. Th. Ottmann

next optimal vertex
Next optimal vertex

fc(x) = c1x1 + c2x2

Ci-1

vi-1

Computational Geometry

Prof. Dr. Th. Ottmann

algorithm 2d lp
Algorithm 2D-LP

Input: 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

thenreturn UnboundedLP(H, c )

else report h1 := h; h2 := h´ ; v2 := l1 l2

let h3,...,hn be the remaining half-planes of H

for i:= 3 to n do

if vi-1  hi then vi := vi-1

else Si-1 := Hi-1 * li

vi := 1-dim-LP(Si-1, c )

if vi not exists then return 

return vn

Running time: O(n²)

Computational Geometry

Prof. Dr. Th. Ottmann

algorithm 1d lp
Algorithm 1D-LP

Computational Geometry

Prof. Dr. Th. Ottmann

algorithm 1d lp11
Algorithm 1D-LP
  • Find the point x on li that maximizes cx , 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-1else
  • return 
    • Time: O(i)

Computational Geometry

Prof. Dr. Th. Ottmann

addition of halfplanes in different orders

Bad

11

12

9

10

7

8

5

6

3

4

1

2

vi

Good

1

2

3

4

5

6

7

8

9

10

11

12

Addition of halfplanes in different orders

Computational Geometry

Prof. Dr. Th. Ottmann

optimal vertex13

h4

h1

h3

v3= v4

h2

h4

h1

h3

c

h5

c

h2

v4

v5

Optimal vertex

Computational Geometry

Prof. Dr. Th. Ottmann

algorithm 2d lp14
Algorithm 2D-LP

Input: 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

thenreturn UnboundedLP(H, c )

else report h1 := h; h2 := h´ ; v2 := l1 l2

let h3,...,hn be the remaining half-planes of H

for i:= 3 to n do

if vi-1  hi then vi:=vi-1

else Si-1:= Hi-1 * li

vi:= 1-dim-LP(Si-1, c )

if vi not exists then return 

return vn

Running time: O(n²)

Computational Geometry

Prof. Dr. Th. Ottmann

addition of halfplanes in different orders15

Bad

11

12

9

10

7

8

5

6

3

4

1

2

vi

Good

1

2

3

4

5

6

7

8

9

10

11

12

Addition of halfplanes in different orders

Computational Geometry

Prof. Dr. Th. Ottmann

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