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subproblem. exchanges the information among the subproblems and tries to reach the prescribed tolerance between the best upper bound issued by sp: and the best lower bound issued by sp:. - Before the algonthm starts. the user selects either the lower bound subproblem, or the upper bound subproblem, from each pan p l. ... N, while ensuring chat at least one subproblem is the lower bound type. and at Ieast one is the upper bound type. DO IN PARALLEL frocessor t for t= 1, ..., N Step O. Set k= 1, E*. and determine whether P is infeasible. - solve spi ; if it is infeasible. send a stop signal to SP, for al1 s+t and stop. P is infesi ble; - if a stop signai from SP, for ail s#t is received, stop, P is infeasible; I I - othenvise. record optimal prima1 and dual solutions x, . y, and nt'. wll. Step 1. Set h=k+L. exchange information, modify SP, and solve it. - set x:" = (x:". xrl) and Y:-' = (Y:*'. y:'): - set n:-' =(n,';-' ', rr:" ') ' and Q:-' =(RF k-17 T, ' , - send (c,r:" + dry:-'), L ~~~F' for dl i. and ~,r:'l to al1 lower bound subproblems; - send K;*'D,, O:*'L~ for al1 i, and (n:-'b, + o:'~$) to dl upper bound subproblems; k- I - if t is a lower bound subproblem. receive (c,r/' + dsys ), L,~:' and from SP, for dl i and ~ t :
- if t is an upper bound subproblem. receive ir,"'~,. u:'L~~~ and (n;-lb, + u:-%) from SP, for dl i and ~ t ; k k k - solve SP: : record optimal z, , xt . yt , and n:, o:; record optimal v:, A:' if t is a lower bound subproblem, and p:, pi"' if r is an upper bound subproblem for dl s and i#t; - broadcast the optimd :: to SP, for al1 ntt; receive the optimal ;k from SP, for ail s+t. - - Step 7. Test for convergence: select the best upper bound. 5 from SP~ , and the best lower bound. - 3 from SP:. - - if (:y - cf) I E - , then go to step 3: - - - i - if (; - - 2) > E and - :,S = - ,-t for dl lower bound subproblems and ci = r: for al1 upper bound subproblems in three consecutive itentions, go CO step 3; - othenvise. go to step 1. Step 3. Calculate the optimal (or feasible) primal and dud solutions, and terminate the algori thm. - If FI, broadcast A,:-' to SP, for al1 sr: if t=u. broadcast to SP, for al1 *t; - receive A,:-' from SPI for wl and p,"' from SP, for td; - if r=I. cdculate the optimal (or feasible) primal and dud solutions for part t - if eu. calculate the optimal (or feasible) primai and dual solutions for part t
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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
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: with proposals or cuts from other s
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
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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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