An Introduction to Stochastic Programming: The Recourse Problem

Transcription

1 An Inroducion o Sochasic Programming: he Recourse Problem George Danzig and Phil Wolfe Ellis Johnson, Roger Wes, Dick Cole, and Me John Birge Where o look in he ex pp. 6-7, Secion.2.: Inroducion o sochasic programming. pp , Chaper 6: heory and algorihms pp , Chapers 7 & 8: Applicaions: Value-a-risk, condiional value-a-risk, asse/liabiliy managemen, corporae deb managemen, synheic opions. Sochasic Programming and GAMS SOCHASIC PROGRAMMING MODELS IN GAMS: Name Descripion ype No. Auhor airsp Aircraf Allocaion OSLSE 89 Danzig, G B airsp2 Aircraf Allocaion - sochasic opimizaion wih DECIS DECIS 96 Danzig, G B aplp Sochasic Programming Example for DECIS DECIS 97 Infanger, G aplpca Sochasic Programming Example for DECIS DECIS 98 Infanger, G clearlak Scenario Reducion: ClearLake exercise LP 249 Birge, J R farm he Farmer's Problem formulaed for DECIS DECIS 99 Birge, J R kand Sochasic Programming OSLSE 87 Kall, P lands Opimal Invesmen OSLSE 88 Louveaux, F V prodsp Sochasic Programming Example OSLSE 86 King, A J prodsp2 Sochasic Programming Example - reformulaed for DECIS DECIS 200 King, A J srkandw Sochasic Programming Scenario Reducion LP 248 Kall, P

2 Anicipaive and Adapive Decisions We can make decisions before random evens, here and now. Or we can wai unil he random evens are revealed, wai and see. hese are called anicipaive, and adapive decisions respecively. Models Wih Only Anicipaive Decisions Models wih only anicipaive decisions are called probabilisic problems (nowadays); originally hey were called chance consrained problems by Charnes and Cooper, [959]. he one-year Aun s Loan Problem is an example. Recourse Model ~ Boh Anicipaive and Adapive Decisions wo Sage Sochasic Linear Program wih Recourse x a x + Eξ[ y ( ξ ) c( ξ) y( ξ)] Ax = b B( ξ) x C( ξ) y( ξ) = d( ξ), y( ξ) 0 Afer We Make Our Firs Sage Decision We Solve, for Each ξ Ξ: We Le: f( x, ξ) = c( ξ) y( ξ) Y ( ξ ) C( ξ) y( ξ) = d( ξ) B( ξ) x y ( ξ) 0 f( x) = E [ f( x, ξ)] ξ 2

3 Our Sochasic Problem Becomes a Deerminisic Non-Linear Problem ax + f ( x ) Wha is f(x) like? If x is finie, f is piecewise linear and concave. When he daa is described by absoluely coninuous probabiliy densiies wih finie second momens, hen f is differeniable and concave. In any case he consrains are linear. Nex o quadraic programming, hese problems are probably he simples class of non-linear problems o deal wih. We Assume Ξ Finie We hen Have: We will limi ourselves o he case where Ξ is finie. Specifically, we le Ξ = {ξ, ξ 2,..., ξ s }, and p = {p, p 2,..., p s } be he probabiliies of each oucome. he ξ s are also called scenarios. S ξ y( ξ) ξ ξ = k y( ξ) ξk ξk k = E [ c( ) y( )] p c( ) Y( ) he Deerminisic Equivalen xy, ( ξ ) a x + pc ( ξ) y( ξ) pc ( ξs) y( ξs) Ax = b B( ξ ) x + C( ξ ) y( ξ ) = d( ξ ) B( ξs) x + C( ξs) y( ξs) = d( ξs) x, y( ξ ),..., y( ξ ) 0 S he L- AKA Bender s Decomposiion 3

4 A Quick Skech of he L- Noe, in he deerminisic equivalen, excep for he B(ξ i ) columns, he problem decomposes ino independen smaller problems. ha is, afer he firs sage decision x is made; he problem decomposes ino independen smaller problems. A Quick Skech of he L- Generally speaking he L- ries o represen he effec of he second sage decisions by consrains on he firs sage by making ouer approximaions of he se of feasible x s and of he 2nd sage conribuion, f(x). We can rewrie his as: ax + f ( x ) ax + θ f( x) θ A Quick Skech of he L- he L- is an ieraive algorihm alernaing beween solving maser programs, and subproblems. he Maser Problem Looks Like: ax + θ Dx d =,..., r Ex+ θ e =,..., s 4

5 A Quick Skech of he L- Where he inequaliies involving D, d represen consrains on x, implied by he inabiliy of he recourse funcion o achieve feasibiliy. he inequaliies involving E,e represen a linearized over-approximaion o he 2nd sage objecive funcion, f(x). hese inequaliies are obained by solving in y(ξ) he subproblems of he deerminisic equivalen for a fixed value of x given by he las soluion of he maser problem. n-sage Recourse Problems Boh mehods generalize o n-sage problems for n 2. We illusrae wih a generalizaion of he Aun s Loan Problem. DECIS OSL OSLSE GAMS Solvers Large scale sochasic programming solver from Sanford Universiy (Infanger) High performance LP/MIP solver from IBM (No suppored by IBM afer 2004) (John Forres, now working wih COINS-OR) OSL Sochasic Exensions for solving sochasic models (No suppored by IBM afer 2004)(Alan King) Yu. Ermoliev & R.J-B Wes (eds.), Numerical echniques for Sochasic Opimizaion, Springer, 988 (Alan King, Ch. 30). John R. Birge & François Louveaux, Inroducion o Sochasic Programming, Springer, 997 (Chapers 5-8). Gerd Infanger, Planning Under Uncerainy, Boyd &Fraser, 994 (Chaper 2). (con.) (con.) Peer Kall & Sein W. Wallace, Sochasic Programming, John Wiley & Sons, 994 (Chaper 3). (Updaed version available on line for free: hp://home,himolde.no/~wallace/ ) R. Van Slyke & R.J-B Wes, L-Shaped Linear Programs wih Applicaion o Opimal Conrol and Sochasic Programming, SIAM J. on Applied Mahemaics, 7, pp , 969. Sochasic Programming Communiy Home Page (hp://soprog.org/ ) hese noes are based on he books by Birge & Louveaux, and Cornuejols & üüncü. 5