Recent Developments in Disjunctive Programming

Transcription

1 Recent Developments n Dsjunctve Programmng Aldo Vecchett (*) and Ignaco E. Grossmann (**) (*) INGAR Insttuto de Desarrollo Dseño Unversdad Tecnologca Naconal Santa Fe Argentna e-mal: aldovec@alpha.arcrde.edu.ar (**) Department of Chemcal Engneerng, Carnege Mellon Unverst, Pttsburgh, PA 53, - USA e-mal: grossmann@cmu.edu

2 Introducton Accepted Formulatons Formulatons MILP MINLP are the most used n the academa and companes to solve problems wth dscrete decsons Algorthms for MINLP - Outer Appromaton (OA) Dcopt ++ - Branch & Bound (B&B) - Generalzaed Benders Decomposton (GBD) - Etended Cuttng Plane (ECP) Modellng Envronments Modelng envronments used for posng and solvng a MINLP problem: GAMS, AMPL, other

3 Alternatve Formulaton to MINLP Hbrd Formulaton (Dsjunctons and 0- varables) mn Z = Σ k c k + f() + d T s.t.: g() 0 r() + D 0 A a Y k h k () 0 k SD D k c k = γ k Ω (Y) = True R n, {0,} q Y {True, False} m, c k 0 ❹ c k are contnuous varables ❹ (0-) are varables ❹ Y k are Boolean varables to establsh whether the dsjunctve term s true or not ❹ f() objectve functon ❹d T lnear cost terms ❹ g() constrants that are ndependent of the dscrete decsons ❹ r()+d 0 med-nteger constrants ❹ A a nteger constrants ❹Ω (Y) propostonal logc relatng Boolean varables

4 Dsjunctve Formulaton From Hbrd formulaton mn Z = Σ k c k + f() s. t.: g() 0 r() + D 0 A a + d T Y k h k () 0 k SD D k c k = γ k Ω(Y) = True R n, {0,} q, Y {True, False} m, c k 0

5 MINLP formulaton From Hbrd formulaton mn Z = Σ f() + d T k c k + s.t.: g() 0 r() + D 0 A a Y k h k () 0 k SD D k c k = γ k Ω(Y) = True R n, {0,} q Y {True, False} m, c k 0

6 Relaatons for a dsjunctve set Bg-M relaaton Lnear case Non-lnear case F = D T n [ a b ] R F = D n [ h () 0] R T a b + M ( - ) h () M (- ) = = M = ma{ a T b lo up } M = ma{ h () lo up }

7 Relaatons for a dsjunctve set Conve Hull up D n D v v 0 D,,0 0 ) / h ( v R,v 0 v - = = v v 0 D,,0 0 b - v a R v, 0 v - up D T n D = = [ ] n T D R b a F = Lnear case Non lnear case [ ] n D R 0 () h F = (Balas, 985) (Lee Grossmann, 999)

8 Relaatons for a dsjunctve set Propertes Increased number of varables and constrants - + Bg-M Conve Hull The Bg-M relaaton of a dsjunctve set mples a med-nteger formulaton of the dscrete decson The Conve-Hull relaaton s related to a dsjunctve formulaton of the dscrete decson Propert The Conve-Hull of a dsjunctve set provdes a relaaton that s tghter or equal to the Bg-M model

9 Cove Hull vs. Bg-M Illustratve eamples X 4 Bg-M Conve Hull X X The Conve Hull s tghtest 3 4 Both relaatons can be equvalent X X F.O. Bg-M Conve Hull 3 4 X

10 Language for dsjunctons and logc propostons Operators, Operands and Sentences Booleans Varables: true and false values, bnares can be used nstead Logc Operators: ( and ), ( or ), ( eclusve or ), ~( not,!, ), ->(mplcaton),<->(equvalence) Selecton Sentences (condtonals) for epressng dsjunctons. We propose IF..THEN..ELSE..ENDIF sentences. Specal sentences to facltate the epressons of logc constrants Sentences lke : atmost, eactl or adjacent.

11 Language for the epresson of dsjunctons Two terms dsjuncton True Constrants set False Constrants set IF (logc epresson) THEN Appl constrants set ELSE Appl constrants set ENDIF vt j vt log(s j log(s b b j j - b - b Y j * j * j, j+, j+ ) + b ) + b φ φ, j+, j b vt j j Y b = 0, j j + = 0 IF (Y j ) THEN t log(s ELSE ENDIF b b j vt j j vt b j j j - b - b b * j log(s, j+, j+ = 0, j + φ ) + b * j φ = 0 ) + b, j+, j

12 Language for the epresson of dsjunctons Several terms dsjunctons Constrants set Constrants set N Constrants set N IF (Logc epresson ) THEN Appl constrants set when logc epresson s true ELSE IF (Logc epresson ) THEN Appl constrants set when logc epresson s true ELSE IF (Logc epresson 3) THEN... ELSE IF (Logc epresson N) THEN Appl constrants set N when logc epresson N s true ENDIF

13 Language for the epresson of dsjunctons Eample: Several terms dsjunctons.5 ) ( ) ( Y ) ( ) ( Y. ) ( ) ( Y IF (Y ) THEN 0.5 ) ( 4) ( + ELSE IF ( Y ) THEN 4) ( 3) ( + ELSE IF ( Y ) THEN.5 ) ( ) ( + ENDIF

14 Language for the epresson of logc propostons Logc operators : ( and ), ( or ), ( eclusve or ), ~( not,!, ), ->(mplcaton), <->(equvalence) for the epresson of relatonshps between the dscrete varables Eample: Y 8 ->Y 3 Y 5 ( Y 3 Y 5 ) Y Y Y 4 Y 5 Y 6 Y 7 The modeler formulates these relatonshps that are translated nto nteger constrants

15 Language for the epresson of logc constrants Specal sentences atmost (parameter lst) Atmost one component of the parameter lst must be true atleast (parameter lst) Atleast one component of the parameter lst must be true eactl (parameter lst) Onl one component of the parameter lst must be true adjacent (lsta de argumentos) Two adjacent components of the parameter lst must be true alldfferent (parameter lst) All components of the parameter lst must take dfferent values

16 Language for the epresson of logc constrants Illustratve eamples atmost (Y()) eactl (Y, Y, Y6, Y8) atleast (Y(), Y, Y6, Y8) eactl (Y, Y, Y6, Y8) atleast (Y(), Y, Y6, Y8) alldfferent (n())

17 Models and Algorthms HYBRID/DISJUNCTIVE PROBLEM Model Specal two terms dsjunctons Reformulate the problem b Bg-M or the Conve-Hull MINLP PROBLEM Model Logc- Based OA B&B based on Conve Hull B&B OA GBD ECP

18 Reformulaton of a dsjunctve problem nto a MINLP Dsjunctve Formulaton MINLP Formulaton mn Z = Σ k c k + f() s. t.: g() 0 Dk h k c Y () 0 k k = γ k k SD Ω(Y) = True R n, Y k {True, False} m, c k 0 mn Z = s. t.: g() 0 = 0 ν h k k A a, ν h k D k k k ( ν k ν D k k k () M =,, / 0, k k k k U γ k k + f() k SD ) 0, k ( -, k k SD ), D D, k SD, k SD {0,}, SD, SD SD k k D k, k SD

19 LOGMIP Implementaton LOGMIP INPUT FILE Precompler Precompler step step GAMS INPUT FILE GAMS GAMS COMPILER COMPILER Matr and control fles Logc Informaton fle Solvers Solvers Soluton

20 LOGMIP Implementaton Precompler The mplementaton mples the epanson of the capabltes of a mathematcal programmng at the level of modelng and soluton technques for non-lnear dscrete problems The objectve of the precompler step s: check the snta and semantcs for dsjunctons, logc propostons logc constrants, specal sentences obtan the logc nformaton and record nto a fle generate a fle read to be compled b GAMS

21 LOGMIP Implementaton Algorthms Implemented Formulaton Algorthms OA/ER/AP : Dcopt++ MINLP B&B : SBB ECP : Prototpe Dsjunctve Logc-Based OA (for specal two terms dsjunctons) Hbrd Logc-Based OA etended (for specal two terms dsjunctons)

22 LOGMIP Implementaton Problem solvng If the model s an MINLP B default we appl (DICOPT++) Alternatves SBB or ECP If the model s a hbrd/dsjunctve program We reformulate the problem as MINLP and proceed Alternatve: Logc-Based OA for specal two terms dsjunctons

23 8 Process Superstructure 3 4 Y Y Y 6 Y 4 Y 5 8 Y Y 7 7 Y

24 Results : 8 Process superstructure Model Algorthm Relaed optmum Iteratons Varab. Constr. Dscrete V. Dcopt NLP 4 major MINLP ECP -5.3* 6 MIP GDB 5.08 NLP 4 major MINLP (Conve hull) Dcopt NLP major Dsjunct. Logc-Based OA NLP major Hbrd Logc-Based OA et NLP major Optmum 68.09

25 Results: predcton of nfrared spectroscop parameters mn Z = w j + k k w j = k {[( c kj - p k a j ) T * R Dsuncones ] *( k c kj - p k a j )] } j = wave number(0) j= epermental data (8) k= components (3) P mn k Y Pk P c = k k ma k Y P c k k k = 0 = 0 Model MINLP Dsjunctve Constrants 39 0 Varables Dscrete Var Optmum CPU Tme 55 sec 7 sec Iteratons 6 4

26 Results: Multproduct Batch Plant Desgn PRODUCTS A,B,C, D, E STAGES Staorage Tank Model MINLP Hbrd Constrants Varables 3 Dscrete varables Obj. value CPU Tme 87 sec. 80 sec. Iteratons NLP 0 tera NLP 4 tera

27 Results: HDA processsnthess Reccle Toluene Toluene Reccle H H Toluene Model MINLP Dsjunct. Constrants Varables 7 77 Dscrete 3 4 Varables Obj. value CPU tme 93 sec. 80 sec. Iteratons NLP tera. NLP tera. Methane Benzene Dphenl

28 Conclusons ❹ Hbrd and Dsjunctve Programmng provde advantages n modelng and soluton technques that complements Med Integer Non Lnear Programmng (MINLP) ❹ We propose a language for the epresson of dsjunctons and logc propostons to etend the mathematcal modelng languages ❹ Startng wth a hbrd/dsjunctve model we propose to reformulate t to MINLP and then solve wth an standard algorthm. ❹ Objectve of ths approach s to gve the modeler several alternatves for modelng and solvng a contnuous/dscrete non-lnear program problem