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CHAPTER 1 introduction TABLE OF CONTENTS 1.1 Brief History of Decomposition ............................................................. ........... (1) 1.2 Motivation and Objectives of the Research .................................................... (3) 1.3 Overview of the Thesis ....................................................................................... (5) CH.4PTER 2 Literature Review .......................................................................................... (7) 1.1 An Overview of the Decomposition Methods .................................................... (7) Dantzig-Wolfe Decomposition Method ........................................................ (7) The Nested Dual Decomposition Method for Multi-Stage Models .............. (8) Benden Decomposition Method ................................................................... (9) The Nested Mmal Decomposition Method for Multi-Stage Models ........... (9) Cross Decomposition Method by Van Roy ................................................... (9) Pnmai-Gud Decomposition Method by Lan and Fuller ............................ (10) ........ Primai-Dual Nested Decomposition Method for Multi-S tage Models (11) 2.2 An Overview of Parallei Computing .............................................................. (13) SIMD vs MIMD ......................................................................................... (13) S hared Memory vs Message Passing .......................................................... (13) Performance Measurement in Paralle1 Computing ..................................... (15) ................................................................................... 2.3 PanIlel Decomposition (16) 2.4 Summq .......................................................................................................... (18) ...................................... CH,- 3 PanIlel Decomposition of the Two-Part Roblern (19) 3.1 Mode1 Structure and Assumption .............................................................. (19) 3.2 Subproblem Structures ..................................................................................... (22) 3.3 The Decomposition Algori thm ........................... . ...................................... (27) 3.4 Properties of the Alprithm .............................................................................. (30) viii
3.5 Sumary and Observations on the Algorithm ................................................. (43) CHAPTER 4 ParaIlel Decomposition of the Multi-Pan Mode1 ........................................ (45) 4.1 Definition of a Multi-Part Linear Program ...................................................... (45) 4.3 The Structure of Subproblems for the Fint Method ........................................ (47) 3.2.1 The Bifurcation Process .......................................................... (47) 4.32 Precise Description of the Subproblems ....................................... (53) 1.2.3 A Strategy to use more Information from Second Level Iteration ........... (62) 4.3 The Parallel Decomposition Algorithm for the First Method .......................... (62) 4.4 Propenies of the Algonthm for the First Method ............................................. (69) 4.5 A Heunstic Decomposition Algorithm - the Second Method .......................... (74) 1.5.1 The Structure of Subproblems for the Second Method .......................... (74) 4.5.2 The Heuristic Decomposition Algorithm for the Second Method ......... (79) 45.3 Properties of the Algorithm for the Second Method .............................. (82) C-R 5 Prelirninary hplementation and Results ................................................... (87) 5.1 The hplementation Procedure ...................................................................... (87) 5.1.1 Decomposition Phase .............................................................................. (89) 5.1.2 Solution Phase ......................................................................................... (94) 5.1 The Test Roblems and Results of the Expriment ......................................... (95) CHAPTER 6 Conclusions and Future Research ............................................................. (107) 6.1 Conclusions ................... ,. ....................................................................... ( 107) 6.2 Future Research ............................................................................................ (1 1 1) Appendix A GAIS codes of the test models ........................................................... (1 13) .\ . I Hydro-Electric Genention Planning Model 1 (HEPGL) ................................ (1 13) A.2 Hydro-Electric Genention Planning Mode1 2 (HEEG?) ...................... . (1 16) A.3 Financial Planning Mode1 (FP) ...................................................................... (120)
Page 1 and 2: A PARALLEL PRIMAL-DUAL DECOMPOSITIO
Page 3 and 4: The University of Waterloo requires
Page 5 and 6: WATPAR, and in each of the tests, t
Page 7: To Soyoung and Katherine Eugene Par
Page 11 and 12: Table 4 . I Table 5.1 Table 5.2 Tab
Page 13 and 14: Chapter 1 Introduction 1.1 Brief Hi
Page 15 and 16: 1.2 Motivation and Objectives of th
Page 17 and 18: 2. Pmve and dernonstrate the conver
Page 19 and 20: Chapter 2 Literature Review This ch
Page 21 and 22: Benders komposition Method In the B
Page 23 and 24: in this algorithm, both subproblems
Page 25 and 26: SIMD vs MIMD Panllel computers cm b
Page 27 and 28: apan. In mmy LAN's, communication i
Page 29 and 30: angular linear programs by fixing t
Page 31 and 32: Chapter 3 Paralle1 Decomposition of
Page 33 and 34: ound constraints. The two parts are
Page 35 and 36: constnicted in the sarne way by res
Page 37 and 38: 1T 17 t-IT T nlk*'T )' and &'-' = (
Page 39 and 40: a proposal is passed back to the fi
Page 41 and 42: END - solve ; if it is infeasible o
Page 43 and 44: l, and Theorem 3. lb mles out the p
Page 45 and 46: nonlinking constraints and upper bo
Page 47 and 48: (The "only if* part) If P is infeas
Page 49 and 50: . - Thmm 3.6 Suppose (-ri. )Jik, VI
Page 51 and 52: the optimal value has ken reached.
Page 53 and 54: spfl, we have have k .J< k PI -1 -
Page 55 and 56: zi - $' 5 z+ - &J-[, and by Theorem
Page 57 and 58: Chapter 4 Paraiiel Decomposition of
Page 59 and 60:
upper bound constraints and nonnega
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type subproblem. The lower bound su
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An algorithm could be defined to ex
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Figure 43 9-stage decomposition pri
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combinations of known solutions of
Page 69 and 70:
lower bound subproblem (parts 1 and
Page 71 and 72:
&.',* and CI::;, are a 1 x (i - 1)
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from the same aggregated subproblem
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problems is discussed. Various prop
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Processor 2 Step O. Set level I cou
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- update and solve spi ; record opt
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for the optimum by exchanging the p
Page 83 and 84:
The next result guarantees that in
Page 85 and 86:
fonn a nondecreasing senes of lower
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one upper bound subproblem. The alg
Page 89 and 90:
su, , for al1 i Note that when k= 1
Page 91 and 92:
with proposals or cuts from other s
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- if t is an upper bound subproblem
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Proof : The Assumption guarantees t
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nNk-'. Proof : For i= 1.2. ... . N
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Chapter 5 Preliminary Implementatio
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5.1.1 Decomposition Phase In the de
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NB,~,PEI 8 # Selectmg the rows ROWS
Page 105 and 106:
Figure 53 Example of multi-part str
Page 107 and 108:
in each computer as discussed in Ch
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followiag 4 parts: scenarios 1 to 1
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"WSET' is the time taken on the RSI
Page 113 and 114:
esults of the parallel decompositio
Page 115 and 116:
Table 5.6 Performance of Parailel D
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We dso anaiyzed the speed-ups and e
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Chapter 6 Conclusions and Future Re
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8. tested several models for conver
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computational speed for some real w
Page 125 and 126:
APPENDIX A GAMS codes of the test m
Page 127 and 128:
hydrA 25, ...., hydrZ 1s /; paramet
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NwatBal(hydro,u,pericds,senarios1 .
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hydrEI 1100, hydrB 1200, ..... hydr
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VARIABLES En 'Expectation of the ut
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APPENDIX B The C codes of WSET We p
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k denoces an index of SubProblern t
Page 139 and 140:
SPb~[kl-+; 1 else ( if (rowinfoIi]
Page 141 and 142:
* Pack initial &ta and send '/ for(
Page 143 and 144:
Appendiv C C codes of WATPAR We pre
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if (LpSub [k] ->rwnmbs [pl > HISPb-
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* Use advanced basis at each iterat
Page 149 and 150:
status = CPXnewcols (env,lp, 1, &Lp
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TERMINATE : ..,. Free up the proble
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short SolveProb(C3XEWptr env, CP.XL
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variables and add other part's link
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i* Add a cut for upper level iterat
Page 159 and 160:
C3 Proecssor 3 : Lower-Upper Bound
Page 161 and 162:
t for(k=l; k c icer-count; k+-1 sta
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{ gectimeofday(&tvl, (struct cimezo
Page 165 and 166:
i status = CPXchgcoef(env, lp, (int
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status = AddLamcols ( LpSub, env, I
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BIBLIOGRAPHY Aardal. K. And A. An.
Page 171 and 172:
Computation". Mathematical Programm
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