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The expression on the right is a sum of products of two vecton. therefore Since ol and have upper bounds of MI and Mz, we can let QI Then. since M2 >O, it follows that Q >O. Therefot-e. = ma Ildz - a l f,lzll, OS-t0 ar j+ . Rwfi For any panicular value of jX1, there is an earlier iteration number, from the subsequence. {k, }, . which is closest to j, i.e. k,, = max (k, 1 k,, I j. je N } . By Theorem 3.5.
zi - $' 5 z+ - &J-[, and by Theorem 3.8, the right side of the inequality converges to zero. Therefore, (z/ - :)'')+O as j- through je N. Cordlary if the parallel primnl-dual decomposition algorithm proceeds to iterution k> 1, rhen with the given tolerance E > O. it stops in a finite number of iterations. RooE The corollary follows directly from the stopping criterion uO and Theorem 3.9. 3.5 Summary and Observations on !Le Algorithm L. The onginai problem is divided into two types of subproblems in each part: a lower bound type subproblem and an upper bound type subproblem. 2. The lower bound subproblem in primai form is consuucted by resaicting prima1 variables. xz and y, to convex combinations of known values. received from the upper bound subproblern. thus dropping out redundani prima1 nonlinking conscraints and primai upper bound constraints. nie upper bound subproblem in dud form is consuucted by restncting dual variables. iri and al, to convex combinations of known values. received from the first subproblem. then dropping out redundant dud nonlinking constraints and dual upper bound constraints. 3. The panIlel aigorithm ha a perfectly bdanced structure by the two master-like subproblems instead of the master and subproblem structure as in the aaditional decomposition methods. 4 information on primai and duai solutions is sent and received between the two subproblems at each iteration. The duai solution of the fint subproblem is passed to
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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 and 52: the optimal value has ken reached.
Page 53: spfl, we have have k .J< k PI -1 -
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
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short SolveProb(C3XEWptr env, CP.XL
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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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