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Table S. 1 presents the statistics of the five test problems. The Linking rows and columns are those constraints and variables which have nonzero enaies in the matrices L, when the original problerns are decomposed into four parts. The parenthesis in "Linking Cols (or "Linking Rows") shows the ratio of the number of linking columns (or rows) to the number of columns (or rows). Problems TEP NAEP #of regions Iscenarios 4 7 Table 5.1 Statistics of the test problems ~olumns 321 3949 ~ows 165 2325 ~iakiag Cols. 120 (37%) 234 (5.9%) Linking Rows Nonzeroes 40 (24%) 841 234 (9.6%) 12033 The linking variables in TEP are 'LFLOWS (R.RR.T)' representing the arnount of electricity that flows from region R to region RR at time T, and the Iinking consaaints are the elecuicity demand consuaints in each region R at time T, 'LTRADE(R,T)'. NAEP has 7 regions of Westem Canada (WC), Eastern Canada (EC), Westem US. 0, Northern U.S. (NU), Eastern U.S. (EU). Southem U.S. (SU) and the Rest of the World (RW). The 7 regions were muped into 4 pans: WC, EC and RW. WU and NU, and EU and SU. The linking variables and C consmints of NAEP are those representing energy flows arnong the 4 parts. HEPGl is fomiulated using rime index and scenario index as shown in Figure 5.5, so it cm be divided into the
followiag 4 parts: scenarios 1 to 12. scenarios 13 to 24. scenarios 25 to 36 and scenarios 37 to 45. HEPGZ has 16 scenarios formulated with nonanticipativity constraints, thus each part having 4 scenarios. FP has 6 periods, each period with 7 scenarios and divided by the first period's scenarios. so the linking variables are the asset variables. 'LXO(A)'. and cash variable, 'LCO*, of pend O since those 5 variables appear in al1 the first period's consnaints. The linking consaaints are the balance consuainü of financial flows in period 1, 'LBALF_L(MS)'. which include the linking vuiables. Figure 5.6 A Scenario me for HEPGI 97
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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
Page 57 and 58: Chapter 4 Paraiiel Decomposition of
Page 59 and 60: upper bound constraints and nonnega
Page 61 and 62: type subproblem. The lower bound su
Page 63 and 64: An algorithm could be defined to ex
Page 65 and 66: Figure 43 9-stage decomposition pri
Page 67 and 68: 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)
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: in each computer as discussed in Ch
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 and 124: 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
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
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C3 Proecssor 3 : Lower-Upper Bound
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t for(k=l; k c icer-count; k+-1 sta
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{ gectimeofday(&tvl, (struct cimezo
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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.
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Computation". Mathematical Programm
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