LP in Standard and Slack Forms

Transcription

1 LP i Stadard ad Slack Fors ax j=1 s.t. j=1 c j a ij b i for i=1, 2,..., 0 for j=1, 2,..., z = 0 j=1 c j x i = b i j=1 a ij for i=1, 2,...,

2 Auxiliary Liear Progra L: LP i stadard for: ax j=1 L aux : Auxiliary LP: s.t. j=1 c j a ij b i for i=1, 2,..., 0 for j=1, 2,..., ax x 0 s.t. j=1 a ij x 0 b i for i=1, 2,..., 0 for j=0, 1, 2,..., L aux is bouded ad feasible.

3 Duality

4 Upper Bouds o Maxiizatio LP ax 4x 1 + x 2 + 5x 3 + 3x 4 s.t. x 1 - x 2 - x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3-5x 4 3 x 1, x 2, x 3, x 4 0 Multiply secod costrait by 5/3: 25 3 x x 2 + 5x x x 1 + x 2 + 5x 3 + 3x x x 2 + 5x x

5 Upper Bouds o Maxiizatio LP ax 4x 1 + x 2 + 5x 3 + 3x 4 s.t. x 1 - x 2 - x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3-5x 4 3 x 1, x 2, x 3, x 4 0 Add the secod costrait to the third costrait: 4x 1 + 3x 2 + 6x 3 + 3x x 1 + x 2 + 5x 3 + 3x 4 4x 1 + 3x 2 + 6x 3 + 3x 4 58

6 Upper Bouds o Maxiizatio LP ax 4x 1 + x 2 + 5x 3 + 3x 4 s.t. x 1 - x 2 - x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3-5x 4 3 x 1, x 2, x 3, x 4 0 Costruct a liear cobiatio of the costraits usig oegative ultipliers y 1, y 2, ad y 3 : y 1 5y 2 y 3 x 1 y 1 y 2 2y 3 x 2 y 1 3y 2 3y 3 x 3 3y 1 8y 2 5y 3 x 4 y 1 55y 2 3y 3 Left-had side will be a upper boud for the LP if the coefficiets at each are at least as big as the correspodig coefficiets i the objective fuctio y 1 5y 2 y 3 4 y 1 y 2 2y 3 1 3y 1 8y 2 5y 3 3 y 1 3y 2 3y 3 5 Ay set of oegative ultipliers y i satisfyig these iequalities also satisfies 4x 1 x 2 5x 3 3x 4 y 1 55y 2 3y 3 Good upper boud: iiize right-had side s.t. costraits.

7 Good Upper Boud i y y 2 + 3y 3 s.t. y 1 + 5y 2 - y 3 4 y 1 + y 2 + 2y 3 1 y 1 + 3y 2 + 3y 3 5 3y 1 + 8y 2-5y 3 3 y 1, y 2, y 3 0

8 Duality The idetificatio of a dual proble is alost always coupled with the discovery of a polyoial-tie algorith. Duality provides a proof that a solutio is optial.

9 LP i Stadard For ad Its Dual ax 3x 1 + x 2 + 2x 3 s.t. x 1 + x 2 + 3x x 1 + 2x 2 + 5x x 1 + x 2 + 2x 3 36 x 1, x 2, x 3 0 i 30y y y 3 s.t. y 1 + 2y 2 + 4y 3 3 y 1 + 2y 2 + y 3 1 3y 1 + 5y 2 + 2y 3 2 y 1, y 2, y 3 0

10 LP i Stadard For ad Its Dual ax j=1 c j s.t. j=1 a ij b i for i=1, 2,..., 0 for j=1, 2,..., i i=1 bi y i s.t. aij i=1 y i c j for j=1,2,..., y i 0 for i=1,2,...,

11 Weak Duality x*: feasible solutio to the prial LP. y*: feasible solutio to the dual LP. Clai Proof: j=1 c j * i=1 bi y i * j=1 c j * j =1 aij i=1 y * * i fro the dual = i=1 j=1 a ij * y i * i=1 bi y i * fro the prial

12 Iportace of Weak Duality x*: feasible solutio to the prial LP. y*: feasible solutio to the dual LP. If j=1 the x* ad y* are optial solutios to the prial ad to the dual LPs, respectively. c j * = i=1 bi y i *

13 Fial Feasible Basic Solutio ad Correspodig Dual Solutio ax z = 28 - x 3 /6 - x 5 /6-2x 6 /3 s.t. x 4 = 18 - x 3 /2 + x 5 /2 + 0x 6 x 2 = 4-8x 3 /3-2x 5 /3 + x 6 /3 x 1 = 8 + x 3 /6 + x 5 /6 - x 6 /3 Basic variables: x 1 =8, x 2 =4, x 4 =18 Objective value z = 28 y i ={ c, i if i N 0 otherwise y 1 = 0 (sice x 4 is basic), y 2 = 1/6, y 3 = 2/3 i 30y y y 3 s.t. y 1 + 2y 2 + 4y 3 3 y 1 + 2y 2 + y 3 1 3y 1 + 5y 2 + 2y 3 2 y 1, y 2, y 3 0

14 Feasible Solutio to the Dual i 30y y y 3 s.t. y 1 + 2y 2 + 4y 3 3 y 1 + 2y 2 + y 3 1 3y 1 + 5y 2 + 2y 3 2 y 1, y 2, y 3 0 y 1 = 0 (sice x 4 is basic) y 2 = 1/6 y 3 = 2/3 Objective value: 30 x x 1/ x 2/3 = 28 1 x x 1/6 + 4 x 2/3 = 3 1 x x 1/6 + 1 x 2/3 = 1 3 x x 1/6 + 2 x 2/3 = 13/6

15 Duality Theore Suppose that SIMPLEX teriates with a feasible basic solutio x* = (x 1 *,x 2 *,..., x *) with: N ad B deotig the obasic ad basic variables for the fial slack for c' deotig the coefficiets of the objective fuctio i the fial slack for. Let y* = (y 1 *, y 2 *,..., y *) be defied by y * i ={ c, i if i N 0 otherwise The x* is a optial solutio to the prial LP, y* is a optial solutio to the dual LP ad j=1 c j * = i=1 bi y i *

16 Proof of Duality Theore We have to show that y* is feasible solutio for the dual, ad j=1 c j * = i=1 bi y i *

17 Proof of Duality Theore Objective fuctio of the fial slack for of the prial is: v * j N c * j =v * j N c * j j B 0 =v * j=1 c * j Optial value of the prial objective fuctio: v * = j=1 c j * j=1 c j =v * j=1 c j * j= 1 c j * =v * j =1 c j * i=1 * yi b i j=1 a ij = v * i=1 bi y * i j =1 c j * i=1 aij y i * This ust hold for every choice of x 1, x 2,..., x. Hece v * = i=1 bi y i * ad c j =c j * i =1 aij y i *, j=1,2,..., Sice c * j 0, k=1,2,...,, we get aij i=1 y * i c j, j=1,2,..., ad y * i 0, i=1,2,...,

18 Prial Dual Cobiatios Dual Optial Ifeasible Ubouded Optial Possible Ipossible Ipossible Prial Ifeasible Ipossible Possible Possible Ubouded Ipossible Possible Ipossible

19 Both Prial ad Dual Ifeasible ax 2x 1 - x 2 s.t. x 1 - x 2 1 x 1 + x 2 2 x 1, x 2 0 i y 1-2y 2 s.t. y 1 - y 2 2 y 1 + y 2 1 y 1, y 2 0

20 Practical Iplicatios If >> the the uber of costrait i the dual will be uch saller tha i the prial. Nuber of pivots i SIMPLEX is usually less tha 1.5 ad oly rarely is higher tha 3. Nuber of pivots icreases very slowly with. Solvig dual will i such cases be ore efficiet.

21 Certificate of Optiality ax j=1 c j s.t. j=1 a ij b i for i=1, 2,..., 0 for j=1, 2,..., i i=1 bi y i s.t. aij i=1 y i c j for j=1,2,..., y i 0 for i=1,2,...,

22 Maxiu Flow Give: A directed graph G = (V,E) where each edge (u,v) E has a real-valued, oegative capacity c(u,v), a source vertex s ad a destiatio vertex t. Fid: A axiu flow f: V V R fro s to t

23 Flow Give: A directed graph G = (V,E) where each edge (u,v) E has a real-valued, oegative capacity c(u,v), a source vertex s ad a destiatio vertex t. A flow fro s to t i G is a real-valued fuctio f: V V R satisfyig: Capacity costraits: f(u,v) c(u,v), u, v V. Skew syetry: f(u,v) = - f(v,u), u, v V. Flow coservatio: Flow value f is defied as v V f s,v

24 Maxiu Flow as LP Proble axiize v V f s,v subject to f(u,v) c(u,v) for all u, v V f(u,v) = - f(v,u) for all u, v V v V f u, v =0 for all u V - { s, t }

25 Maxiu Flow as LP Proble axiize v V x sv subject to x uv c uv for all u, v V x uv = - x vu for all u, v V v V x uv =0 for all u V \ { s, t }