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other thret subproblems for region B, C, and D have the linking (or nonlinking) constrainu and variables defined by similar panem matching rules. Since the primal-dual detomposition algorithm doesn't need any master problem, row master and colurnn master are stated as "notrequired". bbWithgnuplot" shows the following pictures: Figure 5.3 for the original problem with no srnicm extracteci; and Figure 5.1 for the original problem with multi-part structure using WSET. The dark spots indicate the locations of the nonzero elements of the matrîx. Figure 5.1 Example of anonymous matrix generated by GAMS
Figure 53 Example of multi-part structure generated by WSET When the "PD" (PanIlel Decornposition) sobroutine is called, it spawns the executable files in the four PCs by the pvm-spawni) routine from the PVM libnry, asking them to stut their own processes. and then sends each subproblem's data, partitioned by the above schemes, to each machine by the pvm-sendo routine. The consuaint matrix is represented by thRe nonzero vecton: one vector stores nonzero consuaint coefficients by coiumn, another vector has row location numbers of those coefficients and the other vector indicates the row number of the first nonzero element in exh coiumn. However the CPLEX cailabie librvy needs an addibonal vector indicating the number of nonzero elements in each coiumn. so the last vector of CPLEX format should be retrieved from the three vecton of the subproblem in each PC. The data are in the format of column wise vectors. so the colurnns which include that part's nonlinking constraints
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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.
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
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: NB,~,PEI 8 # Selectmg the rows ROWS
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 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
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variables and add other part's link
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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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