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$uWaterloo LaTeX Thesis Template - UWSpace - University of Waterloo$

I Table 6.1 The distinction between Lan's aigorithm and the new parallel decomposition aigorithm Lads algorithm Applied structure / Construction of subproblems 1 One srnall and everything elw / Half and half 1 .- . - - Depth of subproblems Information exchange Weighting scheme Subproblem computation More depth (N- 1 ) Sending nearest neighbor 1 Decornposition algorithm 1 Nested 1 New Parallel algorithm Less depth (floor(10gzN) or l+floor(logzhr)) Broadcasting to othen In this thesis. we have completed several theoreticai and implementation tasks. namely. 1. developed the parailel decomposition algorithm for two-part models, 2. proved the convergence of the rwo-part method as well as other useful properties. 3. developed the parallel multi-part decomposition algorithm, 4. proved several usefui properties of the multi-part method. 5. developed a variant (heuristic) decomposition algorithm for multi-part models. 6. modified SET in order to extnct the multi-pan structure h m GAMS. 7. developed a software for a pdlel primal-duai decomposition soiver using PVM and CPLEX. Weights camied over Serial computation (fonvard md backward) No carryover of weights Parailel computation
8. tested several models for convergence and speedups of the multi-part decomposition aigorithm. The parallel decomposition aigorithm for two-part LPs solves two master-like subproblems. an upper bound subproblem and a lower bound subproblem, simultaneously. Each subproblem works as both the master problem and the subproblem in the naditional decomposition methods: each ûccurnuiates proposais from the other. so is like a master problem yet each contains full details on only its own part. so is like a subproblem. Each subproblem has a balanced structure by having the same amount of primai and dual information and has an equivalent position by conducting the convergence test at the same time. The parailel aiprithm approximates the optimal value of the original problem by calculating a nonincreving upper bound and a nondecreasing lower bound 3t each iteration. This pracedure terminates when the two bounds are considered to be close enough according to the prescribed tolerance. The two-part decomposition principle cm in eetended to more than two parts by applying the decomposition principle recursively in a hienrchical way. Fint, the original problem is divided into an aggregated lower bound subproblem and an aggregated upper bound subproblem. Then. each aggregated subproblem can be again divided into its aggregated lower bound subproblem and its aggregated upper bound subproblem respectively. and this hieranihical decomposing process is repeated until there are no more subproblerns to decompose. in the case of four p m. the original prublem cm have two aggregated subpmblems in the fint level. an aggregated lower bound subproblem (PL) and an aggregated upper bound subproblern (Pu), and in the second level. the PL cm be fiuther decomposed into its lower bound subproblem and its IO9
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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
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upper bound constraints and nonnega
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type subproblem. The lower bound su
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An algorithm could be defined to ex
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Figure 43 9-stage decomposition pri
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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: Chapter 6 Conclusions and Future Re
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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