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

ABSTRACT In this thesis, we develop a new parallel primal-dud decomposition algorithm for multi- part linear programming (LP) problems. We first present a parallel decomposition method for two-part models, in which the two master-like subproblems are solved simultaneously in two different processon and generate an upper bound and a lower bound on the original probiem at each iteration. These two bounds are monotonically improved and converge to within the prescribed toletance of the optimal value by exchanging primai and dual information. Then we extend the basic principles of the two-part aigorithm to multi-part models by applying a hierarchicd decomposition principle recursively. The original multi-part probiem is divided into two aggregated subproblems of lower bound type and upper bound type. and the aggregated subproblems are funher divided into two srnailer aggregated subproblems of upper bound and lower bound. This bifurcation pmcess continues until there are no subpmblems left for futher decomposition. The subproblems are solved in different processon simultaneously and work together to reach an optimal point during the iterations by exchanging prima1 and dud information in the hierarchicd way. Convergence and other usefui properties of the parallei two- part and multi-part algorithms are proven. We developed a parailel decomposition solver for problems of two. three or four parts, called WATPAR (WATerloo PARalell), through the use of GAhlS, the Regex Library, CPLEX 6.0 and PVM (PYallel Virtuai Machine) 3.1 1 on one IBM R SbO workstation and a cluster of four PCs running the Solaris operating system. Several mulu-part LP models are tested using
WATPAR, and in each of the tests, the new parallel decomposition algorithm converged to an optimal value of the original problem in a finite number of iterations. For some large problems, the new method showed some speedups.
Page 1 and 2: A PARALLEL PRIMAL-DUAL DECOMPOSITIO
Page 3: The University of Waterloo requires
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 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
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An algorithm could be defined to ex
Page 65 and 66:
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)
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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
Page 83 and 84:
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
Page 89 and 90:
su, , for al1 i Note that when k= 1
Page 91 and 92:
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
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
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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
Page 113 and 114:
esults of the parallel decompositio
Page 115 and 116:
Table 5.6 Performance of Parailel D
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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
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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
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NwatBal(hydro,u,pericds,senarios1 .
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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
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SPb~[kl-+; 1 else ( if (rowinfoIi]
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* 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
Page 151 and 152:
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
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
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{ gectimeofday(&tvl, (struct cimezo
Page 165 and 166:
i status = CPXchgcoef(env, lp, (int
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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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