Frank Ellison ¹ Suvrajeet Sen ² Yifan Liu ² Gautam Mitra ¹

Transcription

1 Stochastic Decomposition: Motivation, technology and the challenges that it represents Frank Ellison ¹ Suvrajeet Sen ² Yifan Liu ² Gautam Mitra ¹ ¹ Department of Mathematical Sciences Brunel University Uxbridge Middlesex UB8 3PH ² Daniel J.Epstein Dept of ISE University of Southern California

2 OUTLINE Motivation Methodology Data studies Conclusions Technique Methods of Regularisation History Further Research

3 Motivation Sampling approach with ff objectives Approximate solution Avoid full evaluation of ALL scenarios Use Statistical Evaluation Achieve early STOP by use of statistical criteria 3

4 Motivation SLP Paradigm Modelling Simulation Scenario Database SLP Solver 4

5 Motivation SD Paradigm Modelling Simulation Scenarios STOP signal SD Solver 5

6 Motivation SD + SMPS MkII Paradigm Modelling SMPS MkII script SD Solver Simulation 6

7 Methodology The Model:- Linear 2-stage SP 1st Stage:- Z = MIN[c T x + E{Q(x,ω)}] s.t. Ax b 2nd Stage :- Q(x,ω) = MIN[(g) T y] s.t. T ω x + W y h ω 7

8 Methodology General System Flow Take sample(s) build 2 nd stage observation Generate cut as per L-shaped method Solve master, stage 1 problem Apply STOP-criteria quit if OK else repeat Back-testing and Evaluation 8

9 Methodology STOP criteria Re-sample the observation sequence Re-formulate master prob. (do not solve) Calculate Duality Gap in the Incumbent Solution Epsilon test Re-try many (50) times exact 95% success 9

10 Methodology Contrasting SD with SP One sample per iteration in SD Final SD solution is an approximation Final SD solution has only statistical justification SD users get confidence-interval test-results SD employs minimum sampling to get result with pre-set degree of accuracy (statistical) 10

11 Data studies using PT Reg.n Starting seeds are: ,, MODEL NAME:- 4Node, REGULARISATION:- Proximal Term (PT) ITER MEAN CI[LO] CI[HI] INC.EST End of trials - combined results are: Actual optimum:

12 Data studies using PP Reg.n Model Descriptions Stage 1 Stage 2 Scen.os Rows Cols Rows Cols 4Node ,768 Cep LandS ,000,000 YPGP Phone ,768 SSN U/C Storm U/C Pltxp ,650 20Term ,000 Rand ,000 12

13 Data studies using PT Reg.n Model Iter Mean CI(LO) CI(HI) Incumb SLP Max Avg Min Max Est (avg) opt 4Node Cep LandS YPGP Phone SSN Storm Pltxp Term ** Rand ** In the range (251736,255872) 13

14 Conclusions Can get good results with reduced sampling Can directly use continuous distributions BUT statistical variability exacts multiple runs No need to prepare advance scenario database SD a useful tool for simulation studies the proper raison d etre for SD 14

15 TECHNIQUE Stability achieved with two important measures Inner sub-sequence of incumbent solutions Regularise the Master problem 15

16 TECHNIQUE The need for Incumbent Solutions 16

17 TECHNIQUE The need for Regularisation 17

18 TECHNIQUE The need for Regularisation I (incumbent) S (Solution region) O (Optimum) 18

19 METHODS OF REGULARISATION Proximal Term (PT) I (incumbent) S (Solution region) O (Optimum) Higle/Sen (1996) Ruszczynski /Swietanowski(1997) 19

20 METHODS OF REGULARISATION Trust Region (TR) I (incumbent) O (Optimum) S (Solution region) Linderoth/Wright (2003) 20

21 METHODS OF REGULARISATION Level Method (LEV) I (incumbent) O (Optimum) S (Solution region) Lemarechal/Nemirovski/Nesterov (1995) Fabian/Szoke (2007) 21

22 METHODS OF REGULARISATION A-Priori Comparisons TR has advantage LP not QP LEV has substantial disadvantage:- LP used to pre-evaluate the level repeated in STOP-criteria for every trial 22

23 HISTORY 20+-year history see pub.ns of Higle/Sen On the convergence of Algorithms.... (Math.Op.Res 17, 1992) Stochastic Decomposition (Book: Kluwer 1996) Statistical Approximations for Stochastic Linear Programming Problems (An.Op.Res 85, 1999) 23

24 HISTORY Alternative Sampling Methods - SSMO King and Wets (1989) Retrospective optimisation (Healy 1992) Stochastic counter part method (Rubinstein & Shapiro 1993) Sample path optimisation (Plambeck et al 1996) 24

25 Further Research Compare methods of regularisation Extend to include g and W BUT statistical variability exacts multiple runs Other solution methods eg DEQ Extension to SIP 25