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Park, H.J. 1996. "A Primal-Dual Decomposition Method for Multi-stage. Convex Nonlinear Rograms", master's thesis, Department of Management Sciences. University of Waterloo, WaterIoo. ON. CANADA. Rosen, J.B. and R.S. Maier, "Paralle1 Solution of Large-scale, Block-angular Linear Rograrns". Annuls of Ope rations Research 22 ( 1 WO), pp. 23-4 1. Ruszczynski, A. "Parailel Decomposition of Multistage Stochastic Programming Roblems". Mathematical Programming 58 (1993). pp201-128. Van Roy. TJ. "Cross Decomposition For Mixed Intepr Rogramming ". Mathematical Programming 25 (1983). pp. 46-63. Vladimirou. H. "bComputational assessrnent of distributed decomposition methods for stochastic linear programs", European Journal of Operationui Research 108 (1998). pp. 653-670. Zenios. S.A. 1989, "Parailel Numerical Optirniration: Cumnt Status and an Annotated Bibliognphy". ORSA Journal on Computing, Vol. 1, No. 1. ppl0-pp43.
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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
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lower bound subproblem (parts 1 and
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&.',* and CI::;, are a 1 x (i - 1)
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from the same aggregated subproblem
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problems is discussed. Various prop
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Processor 2 Step O. Set level I cou
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- update and solve spi ; record opt
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for the optimum by exchanging the p
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The next result guarantees that in
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fonn a nondecreasing senes of lower
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one upper bound subproblem. The alg
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su, , for al1 i Note that when k= 1
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with proposals or cuts from other s
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- if t is an upper bound subproblem
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Proof : The Assumption guarantees t
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nNk-'. Proof : For i= 1.2. ... . N
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Chapter 5 Preliminary Implementatio
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5.1.1 Decomposition Phase In the de
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NB,~,PEI 8 # Selectmg the rows ROWS
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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
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: Computation". Mathematical Programm
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