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We first solve them directly using the sirnplex method and the barrier method of CPLEX 6.0 on the RSl6ûûû in order to compare the solution of the direct method with that of our decomposition algorithm. We mesure the solution time on the RSf6000 as elapsed time, which includes the mode1 genention time by GAMS and the solver time by CPLEX, in order to give an idea of how long the whole process talces. Note that HEPG2 cannot be solved by the barrier method. due to insufficient memory. and FP is also too large to be solved. due to the memory limit. for both the simplex method and the barrier method. C Table 5.2 The test statistics of direct methods on RS/6000 (Time is rneasured in seconds). f roblems TEP NAEP HEPG L HEPG2 FP I Simplex Method Banier Method I Sol. Time Iter. No. 0bj.Vdues Sol. Time Iter. No. Obj.Values I 0.46 226 102.52 1 2837 435 873 NIA L 33245 55387 N/A 4154.840 -30 16.200 -144008.473 -30307.573 NIA 3 154.840 -30 16.200 We dso solved the original problems in one PC by transfemng al1 data, using WSET, in order to compare the solution time with that of parailel decomposition in the sarne environment. Since the mificial variables are added in the linking consuiiints in this method, the iteration numbers could be different from those in Table 5.2. In Table 5.3, "Simp. Iter." or "Bar. Itef shows the number of itentions using the simplex method or the barrier method of CPLEX and I 0.45 lOJ.45 958 N/A NIA 12 37 33 N/A - 144008.374 N/A NIA NIA
"WSET' is the time taken on the RSI60O for patitioning the original problem into subpmblems after mode1 generation by GAMS; in this case, the= is no partitioning, but WSET is used to pass the problem to CPLEX on the PC. "Setup" in Table 5.3 is the time for receiving subproblem data and loading the problem for the fiat iteration, and "Sol" represents the whole time to solve the problem in one PC, after setup is cornpiete. TEP NAEP HEPG i HEPG2 FP Table 5.3 Statistics of Simplex and Barrier methods on one PC using WSET Simp. Iter. 230 Simplex Solver - - Time Obj. Bar. Value Iter. Barrier SoIver Time Note that there are slight differences between the two methods in 'WSET' and 'Setup' since the WSET time is dependent on the computer and communication network loads, and the Setup is different for the two methods. The following table gives ihe sizes of the subproblems that are actually solved at the first itention of the decornposition algorithm.
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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
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 and 108: in each computer as discussed in Ch
Page 109: followiag 4 parts: scenarios 1 to 1
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
Page 159 and 160: 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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