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The assumption ensures the boundedness of the solutions to the subproblems defined below, which simplifies the algorithm. The assumption is also used in the proof of convergence of the algorithm. Note that the optimal values of the artificial variables are dso bounded because the costly nature of the artificial variables keep hem as close as possible to their lower bounds of zero during the whole optimization process. (This is shown fonnally in the pmof of Theorem 3.8 in section 3.5). 3.2 Subproblern Structures The new decomposition method relies on the definition of a subproblem for each part: the subproblem for one pan gives an upper bound on the optimal value of the original problem and the subproblem for the other part provides a lower bound for the original problem. Because of the symmetry of the pans in the generai problem statement. the choice of subproblem type for a pan appears to be arbitrary, but for a given instance of the two-part LP, the modeller rnq have Rasons to make a panicular choice. in whai follows. the subproblem for part one is of rhe lower bound tm, and part two provides an upper bound. The lower bound subproblem from pan one, denoted by SP: - at iteration k, cm be constructed by restricting the prima1 variable vecton XI and y? to convex combinations of known solutions of - 2 and y2, obtained from the second subproblem at earlier iterations, thus allowing us to drop out the sets of nonlinking constraints and upper bound consaaints of part two fmm the original prirnal problem. The subproblem SP: - keeps both parts' prima1 Linking - constraints. The upper bound subproblem in dual fom frorn part two. denoted by SD~. cm be
constnicted in the sarne way by restncting the dual variable vectors xi and ol to convex combinations of known values of x1 and oi from the fiat subproblem at earlier iterations. thus allowing us to drop out the sets of dual nonlinking constnints and dual upper bound constraints of part one from the original dual problem, but keeping both parts' dual linking constraints. The primai md dual foms of the lower bound subproblem at iteration k are presented (dl S.!. AI .ri + DI y, 5 bi Bi~i+ Li1 Yi - vt + Lir YI A If i (J) - 1 t-1 (P:) (@J Yi Lzi YI + 5 ui 7 k-1 - (B~X:~+L&')A v~ sfL ek" A'.' = 1 xl,Yi,vi.v:.hk" '0 1 ' k- 1 and Y:"' = (Y? . y-. ... . y ), corne from the (k-1) previous primai solutions of SPz; kk-' is a (k-l)-dimensional column vector variable. i.e. ik-' = (A:-', À'*', ... . h-1 ) , is an k-1 T.
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: ound constraints. The two parts are
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 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 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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