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F I ( L?: wi SM,. r=1.1 ..... N ~~.o,*P,>O for t = 1.2. ... , N where the subscripts r and s indicate the part number, N is the total number of parts, x,, y, and v, are the vecton of nt, r,, and q, variables for part t. for t=l. 2, .... N. The dual variable vecton for the constraints of part r of P are denoted by row vecton n,, co, and p, which are the vectors of nt,. q,, and r, variables for pm t. L, is a qtxr, matrix, A, is an m,x n, matrix, Br is a qtx n, mütrix. DI is an m,xrs matrix, and cr, d,, MIS. b,, f,, and ut>O are vectors of suitable dimensions for r=l. 2. .... iV and s=l, 2, ..., N. The general structure of the multi-part mode! is dso shown in Table 1. The sme assumption as in the two-part case is made in order to simplify the algorithm and to gumntee convergence: Assum~tion: For each pan. r=l. 2. ..., N. the set of nonlinking constraints. together with 46
upper bound constraints and nonnegativity constraints, define a bounded feasible region for the x,, y, vecton. ouai 1 Riml Vector Vecior 1 Dimension Ph' 1 N 1 Objective Table 4.1 General structure of multi-part mode1 ni n qt n2 fi q? .... n m q~ RHS 4.2 The Structure of Subproblems for the Fi Method 4.2.1 The Bifurcation Process We fint divide the original multi-part probiem into an aggregated lower bound subproblem. denoted by PL, and an aggregated upper bound subproblem, denoted by Pu, by applyng the basic idea of paralle1 decomposition of two-part models to multi-part models. Agin. we divide the ag-gegated lower bound subproblem into its aggregated lower bound
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A PARALLEL PRIMAL-DUAL DECOMPOSITIO
Page 3 and 4:
The University of Waterloo requires
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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 and 54: spfl, we have have k .J< k PI -1 -
Page 55 and 56: zi - $' 5 z+ - &J-[, and by Theorem
Page 57: Chapter 4 Paraiiel Decomposition of
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
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followiag 4 parts: scenarios 1 to 1
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"WSET' is the time taken on the RSI
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esults of the parallel decompositio
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Table 5.6 Performance of Parailel D
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We dso anaiyzed the speed-ups and e
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Chapter 6 Conclusions and Future Re
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8. tested several models for conver
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computational speed for some real w
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APPENDIX A GAMS codes of the test m
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hydrA 25, ...., hydrZ 1s /; paramet
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NwatBal(hydro,u,pericds,senarios1 .
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hydrEI 1100, hydrB 1200, ..... hydr
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VARIABLES En 'Expectation of the ut
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APPENDIX B The C codes of WSET We p
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k denoces an index of SubProblern t
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SPb~[kl-+; 1 else ( if (rowinfoIi]
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* Pack initial &ta and send '/ for(
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Appendiv C C codes of WATPAR We pre
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if (LpSub [k] ->rwnmbs [pl > HISPb-
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* Use advanced basis at each iterat
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status = CPXnewcols (env,lp, 1, &Lp
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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
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i* Add a cut for upper level iterat
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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.
Page 171 and 172:
Computation". Mathematical Programm
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