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the second subproblem to make a cut and the pnmal solutions of the second subproblem are passed to the fint subproblern as the proposal information. 5. The new aigorithm approximates the objective function value by issuing monotonicaily improved lower bounds and upper bounds by the first subproblem and second subproblem. respectively during the iterations. 6. This algorithm performs the convergence test by refemng to both subproblems simultaneously and converges to a given tolerance in a finite number of steps. - 7. If, in the upper bounding subproblern. spf. al1 cuts except the most ment are eli minated, then the algori thm reduces to the familiar Dantzig- Wolfe decomposition algori thm. 8. K. in the lower bounding subproblern. SP:, - the weight on the most recent proposd is required to equal 1 (and the other weights are zero). then the algorithm reduces to the fmi liv Benders decomposition algorithm. 9. The panllel decomposition algorithm cm be extended to more than two part problems. Chapter 4 discusses a padlel algorithm for the multi-part problems.
Chapter 4 Paraiiel Decomposition of the Multi-Part Mode1 In this chapter, two parallel decomposition methods for multi-part iinear programrning problems are presented. The fint is developed by applying a hierarchical decomposition principle recunively. We first divide the original multi-part problem into two aggregated subproblems of lower bound type and upper bound type, by applying the basic algorithm of the two-part method. which was discussed in the previous chapter. Then. the aggregated subproblems are further divided into two smaller aggregated subproblems of upper bound and lower bound. This results in some primai subproblems accumulating both proposais and cuts. This bifurcation process continues until there are no subproblerns left for funher decomposition. The subproblems are solved in different processon simuItaneously and work together to reach an optimal point during the iterations by exchanging information in the hierarchicd way. In Chapter 5, we repon successhl convergence on several test problems. The second method is less complen than the first. It defines some subproblems to be of the lower bound type from Chapter 3, Le. utilizing primal proposais from other subproblems, and the other subproblems are of the upper bound type. Unfortunately, this method fails to converge in some tests. We include it because it may be useful as a heuristic. 4.1 Definition of a Multi-Part Linear Program The definition of the two-part problems is now extended to that of multi-part problems. Consider the following prima1 and dual forms of LP problem of N parts:
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A PARALLEL PRIMAL-DUAL DECOMPOSITIO
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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 and 54: spfl, we have have k .J< k PI -1 -
Page 55: zi - $' 5 z+ - &J-[, and by Theorem
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
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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.
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Computation". Mathematical Programm
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