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subsequence <strong>of</strong> {&}, . indexed by S2 ç; SI. Proceeding in a sirnikir fashion, we may conclude that there exist subsequences indexed by S=& such that (w:}converges for every i=l. 7. ..., 1. when ES. The inequalities in the assumption hold for any j>k. so they hold in particular for adjacent elements <strong>of</strong> the subsequence S. Iabeled ka. ka+ Because the difference -Lm*/ _ c 2 is bounded below by O and above by an expression which converges to O through the subsequence S. the daim is proven. subsequence S = {k,, }:, E N such thar the objectives converge ro equal values in rhe limit. i.e. Frorn~pi . we have Subtracting (3.4) from (3.5) gives
spfl, we have have k .J< k PI -1 - Pi - Combining (3.1 1) and (3.12) with (3.10) gives Rearm9ng (3.13). and using Theorem 3.5 gives
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 and 8: To Soyoung and Katherine Eugene Par
Page 9 and 10: 3.5 Sumary and Observations on the
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: the optimal value has ken reached.
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 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 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
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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