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Due to some difficulties in coding and time limits. we could not develop the parallel decomposition solver for more than 4 parts. However. it should be extended to handle any number of parts. This could be done with more advanced coding techniques such as the object oriented programming pandigrn. ui its current fonn, the parallel decomposition solver does not calculate optimal prima1 and dual vectors. To be of practicai use to modellen, these should be calculated and sent back to GAMS to allow genention of a solution report. Implementing the parallel algorithm among physically disvibuted computen with local databases for locally developed subproblerns and cornmunicatirtg with one another by emails or web bmwsea would be another interesting subject from a practicai point of view. Needless to say, more testing and implernentations are required to assess the efficiency of the panllel decomposition algorithm for various sized and structured problems.
APPENDIX A GAMS codes of the test models We present the full details of GAMS models such as HEPG1. HEPG2 and FP. The data of HEPGl and HEPG? are genented nndornly. A part of data is aven here to present the data format. A.1 Hydro-Electric Goneration Planning Mode1 1 (HEGP1) Witten by J.R. 9irge and C. Supacgiat, U. Michigan, 1998' tasc rnodified H. Sin Park Sinclude 'columbiaB.dat" Sinclude 'col~iaB.prn" VMIABLSS COST; 2OSITIVE variables Nthe--sen (the,--u, nodes, Ki =he,m production in each mode in each month (MWh) Ngen,flow(hy~o,u,modes.R) hydrc water release for generaïion in each mode Ln each moncn im3 1 Xevel(hydro,u,Kl water Level at ~ h e end of perzod (Md) Nspi~lF~cw(hydro,u,nades,X) wacer spill over the period due CO over capacity (m3) Nexchange(areas.areas,modes.Ki mount power transfer from (1st) to (2nd) in each mode, each node (M'Wh) Nrat-f~ow~hydrro,u,modes,K) slack ic niziznum flow cmstraints. It sets penalty in :he 3biec:ive functron. Xe-xpCa~a 1 NE ord (areas 1 , Nexc~azgeiareas,areas1,aodes,R)i =L= eqmrt(arensi nm-hour(K1 ~LX(F~Z=C~S Sdru'a=io~(modes,K,periods~, durationimodes,K,periods) 1 ; (areas, scdes , Ki S (ûrd (K) ÛT L) . . SUM(areaslS (ord (areas1
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A PARALLEL PRIMAL-DUAL DECOMPOSITIO
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The University of Waterloo requires
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WATPAR, and in each of the tests, t
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To Soyoung and Katherine Eugene Par
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3.5 Sumary and Observations on the
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Table 4 . I Table 5.1 Table 5.2 Tab
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Chapter 1 Introduction 1.1 Brief Hi
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1.2 Motivation and Objectives of th
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2. Pmve and dernonstrate the conver
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Chapter 2 Literature Review This ch
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Benders komposition Method In the B
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in this algorithm, both subproblems
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SIMD vs MIMD Panllel computers cm b
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apan. In mmy LAN's, communication i
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angular linear programs by fixing t
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Chapter 3 Paralle1 Decomposition of
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ound constraints. The two parts are
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constnicted in the sarne way by res
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1T 17 t-IT T nlk*'T )' and &'-' = (
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a proposal is passed back to the fi
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END - solve ; if it is infeasible o
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l, and Theorem 3. lb mles out the p
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nonlinking constraints and upper bo
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(The "only if* part) If P is infeas
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. - Thmm 3.6 Suppose (-ri. )Jik, VI
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the optimal value has ken reached.
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spfl, we have have k .J< k PI -1 -
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zi - $' 5 z+ - &J-[, and by Theorem
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Chapter 4 Paraiiel Decomposition of
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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
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lower bound subproblem (parts 1 and
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&.',* and CI::;, are a 1 x (i - 1)
Page 73 and 74: from the same aggregated subproblem
Page 75 and 76: problems is discussed. Various prop
Page 77 and 78: Processor 2 Step O. Set level I cou
Page 79 and 80: - update and solve spi ; record opt
Page 81 and 82: 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
Page 87 and 88: 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
Page 93 and 94: - if t is an upper bound subproblem
Page 95 and 96: Proof : The Assumption guarantees t
Page 97 and 98: nNk-'. Proof : For i= 1.2. ... . N
Page 99 and 100: Chapter 5 Preliminary Implementatio
Page 101 and 102: 5.1.1 Decomposition Phase In the de
Page 103 and 104: 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
Page 109 and 110: followiag 4 parts: scenarios 1 to 1
Page 111 and 112: "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
Page 117 and 118: We dso anaiyzed the speed-ups and e
Page 119 and 120: Chapter 6 Conclusions and Future Re
Page 121 and 122: 8. tested several models for conver
Page 123: computational speed for some real w
Page 127 and 128: hydrA 25, ...., hydrZ 1s /; paramet
Page 129 and 130: NwatBal(hydro,u,pericds,senarios1 .
Page 131 and 132: hydrEI 1100, hydrB 1200, ..... hydr
Page 133 and 134: VARIABLES En 'Expectation of the ut
Page 135 and 136: APPENDIX B The C codes of WSET We p
Page 137 and 138: 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
Page 145 and 146: if (LpSub [k] ->rwnmbs [pl > HISPb-
Page 147 and 148: * Use advanced basis at each iterat
Page 149 and 150: status = CPXnewcols (env,lp, 1, &Lp
Page 151 and 152: TERMINATE : ..,. Free up the proble
Page 153 and 154: short SolveProb(C3XEWptr env, CP.XL
Page 155 and 156: variables and add other part's link
Page 157 and 158: 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
Page 163 and 164: { gectimeofday(&tvl, (struct cimezo
Page 165 and 166: i status = CPXchgcoef(env, lp, (int
Page 167 and 168: status = AddLamcols ( LpSub, env, I
Page 169 and 170: BIBLIOGRAPHY Aardal. K. And A. An.
Page 171 and 172: Computation". Mathematical Programm
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