Linear programming: complicating constraints

Transcription

1 Recet Developmets i Power Systems Operatio ad Plaig Jue 9 Liear programmig: complicatig costraits Natalia Alguacil Code Área de Igeiería Eléctrica, Ciudad Real Uiversidad de Castilla-La Maca

2 Cotets Itroductio Complicatig Costraits. Problem Structure Decompositio Datzig-Wolfe Decompositio Algoritm Descriptio Bouds Master problem Alterative formulatio of te Master problem Eamples Jue 9 Natalia Alguacil Code

3 Itroductio Cetralized decisio maig E * M E y* E z* Jue 9 Natalia Alguacil Code

4 Itroductio Decetralized decisio maig E M λ E y E z Jue 9 Natalia Alguacil Code 4

5 Itroductio Costraits tat prevet a distributed solutio: subect to subect to subect to Jue 9 Natalia Alguacil Code 5

6 Itroductio Costraits tat prevet a efficiet solutio: Jue 9 Natalia Alguacil Code 6

7 Complicatig Costraits. Problem Structure Cosider te liear programmig problem Miimize c T z Subect to : E f A b up COMPLICATING CONSTRAINTS Jue 9 Natalia Alguacil Code 7

8 Complicatig Costraits. Problem Structure is a decisio variable vector of size up is te upper boud vector of of size c is a cost vector of size z is te obetive fuctio value (scalar) A is a m costrait matri E is a bloc- diagoal q costrait matri. Te umber of blocs is r Jue 9 Natalia Alguacil Code 8

9 Complicatig Costraits. Problem Structure Ab are complicatig costraits Te value of te i-t row of A is deoted as r i Jue 9 Natalia Alguacil Code 9

10 Complicatig Costraits. Problem Structure Alteratively, subect to complicatig costraits Jue 9 Natalia Alguacil Code

11 Complicatig Costraits. Problem Structure Eample: Jue 9 Natalia Alguacil Code

12 Complicatig Costraits. Problem Structure Eample: c c c E f E f E A A A f b Jue 9 Natalia Alguacil Code

13 Jue 9 Natalia Alguacil Code 7 w z z z y y y 4 Complicatig Costraits. Problem Structure Eample:

14 Complicatig Costraits. Problem Structure Relaed problem Miimize c T z Subect to : E f up Jue 9 Natalia Alguacil Code 4

15 Complicatig Costraits. Problem Structure Relaed problem Miimize ;,..., c Subect to : e i f i ; i,..., q up ;,..., Jue 9 Natalia Alguacil Code 5

16 Complicatig Costraits. Problem Structure Relaed problem eample E f E f E f Jue 9 Natalia Alguacil Code 6

17 Jue 9 Natalia Alguacil Code 7 z z z y y y Relaed problem eample Complicatig Costraits. Problem Structure

18 Jue 9 Natalia Alguacil Code 8 Decomposable relaed problem. Subproblem up i i,..., ; q,..., q i ; f e to : Subect c Miimize,..., ; Note tat:, q Complicatig Costraits. Problem Structure

19 Jue 9 Natalia Alguacil Code 9 E f y y y Subproblem eample Complicatig Costraits. Problem Structure

20 Decompositio Relaed problem solutio p solutios are cosidered:, z,..., ; ;,...,p,...,p were z is te -t compoet of solutio te obective fuctio value of solutio Jue 9 Natalia Alguacil Code

21 Decompositio Complicatig costrait evaluatio Te values of te complicatig costraits for te p solutios are r,r,...,rm ;,...,p r i were is te value of te i-t complicatig costrait for solutio Jue 9 Natalia Alguacil Code

22 Decompositio Master weigtig problem Alterative formulatio of te origial problem Miimize u ;,..., p p z u Subect to : p r u p u b : λ : σ ;,..., m u ;,..., p Variables u are weigtig coefficiets for te solutios Jue 9 Natalia Alguacil Code

23 Jue 9 Natalia Alguacil Code Decompositio u p,..., ; u : u u m,..., ; : b u r u r to : Subect u z u z Miimize p p p p,..., ; u u, σ λ Addig a ew basic feasible solutio

24 Jue 9 Natalia Alguacil Code 4 Decompositio Te reduced cost vector of o-basic variables is Particularly, te reduced cost for te additioal variable u is N λ c d T T N T [ ] r r σ...λ λ z d m m M Additioal solutio reduced cost

25 Jue 9 Natalia Alguacil Code 5 Decompositio Te reduced cost is: Taig ito accout tat: Te reduced cost becomes ad m σ r λ z d c z a r m σ a λ c d m σ a λ c d Additioal solutio reduced cost

26 Decompositio Temiimumreducedcostiscomputedas Miimize ;,..., v c m λ a Subect to : e i f i ; i,...,q up ;,..., Note tat costat σ is removed from te obective fuctio Jue 9 Natalia Alguacil Code 6

27 Decompositio Te previous problem is similar to te relaed origial problem Costraits are idetical However, obective fuctio cost coefficiets ave bee modified Jue 9 Natalia Alguacil Code 7

28 Jue 9 Natalia Alguacil Code 8 Decompositio It decomposes i subproblems. Subproblem is: up i i m,..., ;,..., ;,...,q q i ; f e : Subect to a λ c Miimize

29 Decompositio Te miimum reduced cost is: d v σ c Terefore, two posibilities eist: m λ a σ If d ad o cost improvemet is possible, te optimal solutio correspods to curret u s If d < ad a cost improvemet is acieved icludig te solutio wit te weigtig variable u Jue 9 Natalia Alguacil Code 9

30 Decompositio Decompositio structure MASTER PROBLEM λ r * λ SUBPROBLEM r * λ SUBPROBLEM r * λ SUBPROBLEM N- r * N- MIN λ SUBPROBLEM N r * N Jue 9 Natalia Alguacil Code

31 Datzig-Wolfe Algoritm Step. Iitializatio Geerate p solutios of te relaed primal problem (subproblems); tat is, solve p times te problem Miimize ;,..., c Subect to : e i f i ; i,...,q up ;,..., Jue 9 Natalia Alguacil Code

32 Datzig-Wolfe Algoritm Step. Master problem solutio Solve te master problem ad compute te dual variables Miimize u ;,..., p p z u Subect to : p r u p u b : λ : σ ;,..., m u ;,..., p Jue 9 Natalia Alguacil Code

33 Datzig-Wolfe Algoritm Step. Subproblem solutio Modify obective fuctio costs ad solve Miimize ;,..., c m λ a Subect to : e i f i ; i,...,q up ;,..., ad get a ew solutio Jue 9 Natalia Alguacil Code

34 Datzig-Wolfe Algoritm Step. Covergece cecig Compute te reduced cost of te ew solutio d c m λ a σ If d, stop, optimal solutio foud else if d <, iclude te ew solutio i te master problem ad go to Step Jue 9 Natalia Alguacil Code 4

35 Jue 9 Natalia Alguacil Code 5 Bouds At iteratio ν te obective fuctio of te master problem is: A upper boud is: A lower boud is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) m ν ν m ν m m ν ν dow p ν ν up p ν b λ v b λ a λ c z u z z u z ν ν

36 Alterative master problem Jue 9 Natalia Alguacil Code 6

37 Datzig-Wolfe Eamples Miimize Subect to : X,X,X z 7 complicatig costrait Te solutio is: ; ; Jue 9 Natalia Alguacil Code 7

38 Jue 9 Natalia Alguacil Code 8 Datzig-Wolfe Eamples Step. Iitializatio Relaed problems : Subect to Miimize X,X,X first feasible solutio:,, ; z ; r 8 Subect to: Miimize X,X,X oter feasible solutio:,, ; z 7; r

39 Datzig-Wolfe Eamples Step. Master problem solutio Miimize u 7u z u,u Te solutio of tis master problem is ( ) up Subect to: 8u u 7 : λ z λ σ u () () u u : σ u () () ( ) up u 45, u 5 4 Jue 9 Natalia Alguacil Code 9

40 Jue 9 Natalia Alguacil Code 4 Datzig-Wolfe Eamples Obective fuctio ( ) ( ) ( ) ( ) ( ) ( ) m a λ c a λ c a λ c a λ c Step. Subproblem solutio

41 Datzig-Wolfe Eamples Miimize Subect X to : X X,, Te solutio of te above problem is: v (),, ( ) ( ) ( ) lo i i i ( ) ( ) z v λ b 7 Jue 9 Natalia Alguacil Code 4

42 Datzig-Wolfe Eamples Step. Optimality cec d ( σ ) ( 4 4) d d < New solutio sould be icluded i te master problem r z 6 Te algoritm cotiues i Step Jue 9 Natalia Alguacil Code 4

43 Datzig-Wolfe Eamples Step. Master problem solutio Miimize u 7u u z u,u,u ( ) up () u u u : λ () Subect to: u u u : σ u Te solutio of te above problem is: u, u, u λ σ u u () () ( ) up z. 5 Jue 9 Natalia Alguacil Code 4

44 Datzig-Wolfe Eamples Step. Subproblem solutio Miimize Subect to,, : 5 4 Jue 9 Natalia Alguacil Code 44 Te solutio of te above problem is:,, ( ) ( ) ( ) 5 zlo v λi b i i v ( )

45 Jue 9 Natalia Alguacil Code 45 Datzig-Wolfe Eamples Step. Optimality cec 4 5 d 4 d 4 Optimal solutio foud u u u σ 4 5 d 4

46 Datzig-Wolfe Eamples subect to: complicatig costraits Te solutio is: Jue 9 Natalia Alguacil Code 46

47 Datzig-Wolfe Eamples Te solutio is: Jue 9 Natalia Alguacil Code 47

48 Datzig-Wolfe Eamples Step. Iitializatio Relaed problems subect to: subect to: Jue 9 Natalia Alguacil Code 48

49 Datzig-Wolfe Eamples Step. Master problem solutio Te solutio of tis master problem is: Jue 9 Natalia Alguacil Code 49

50 Datzig-Wolfe Eamples Step. Subproblem solutio Jue 9 Natalia Alguacil Code 5

51 Datzig-Wolfe Eamples Step. Optimality cec Te algoritm cotiues i Step Jue 9 Natalia Alguacil Code 5

52 Datzig-Wolfe Eamples Step. Master problem solutio Te solutio of te above problem is: Jue 9 Natalia Alguacil Code 5

53 Datzig-Wolfe Eamples Step. Subproblem solutio Te solutio of te above problem is: Jue 9 Natalia Alguacil Code 5

54 Datzig-Wolfe Eamples Step. Optimality cec Te algoritm cotiues i Step Step. Master problem solutio Te solutio of te above problem is: Jue 9 Natalia Alguacil Code 54

55 Datzig-Wolfe Eamples Step. Subproblem solutio Sice o etreme poit ca be added te algoritm cotiues i Step Step. Optimality cec Optimal solutio foud: Jue 9 Natalia Alguacil Code 55