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

Figure 1.2 Genenl structure of the multi-part models The panllel primaldual decomposition dgorithm for two part problems converged consistently to the optimal solution in a small nurnber of iterations, however. in some tests of multi-part problems the aigorithm (we cd1 this a heuristic parailel aigorithm - set section 4.5) did not converge exactly to m optimal solution but repeated the same feasible solution thereafter. After we found some erron in the convergence proof of this multi-part algorithm, we finaily reached the ideas of applying the two-part decomposition principle recunively in a hierarchical way since the two-part algorithm seemed to be working fine and no flaws could be found in the convergence proof. Hence. the objectives of this Ph.D. resev~h are as follows. 1. Develop a new paralle1 primai-dual decomposition method for NO-part and multi-part linex proCpmming models which have the generai multi-part matrix structure shown above.
2. Pmve and dernonstrate the convergence of the parallel two-part and multi-pan algorithms as well as other properties of the algorithm. 3. Design and implement a parailel primai-duai decomposition solver for the parallel algorithm on several computen and investigate the computational efficiency of the algorithm. 13 Overview of the Thesis This thesis is organized as follows. The next chapter presents a literature ~ view on both serial decomposition methods and parailel &composition methods with basic concepts of parallel computing. Chûpter 3 proposes the new decornposition method for NO-part linear prognms. which divides the original problem into two master-like subproblems, a lower bound subproblem and an upper bound subproblem. and coordinates them to converge to an optimal solution by exchanging primal and dual information. Roofs of the convergence as well as other useful properties of the algorithm are given. Chapter 4 discusses the development of the parallel algorithm for multi-pan models. in which the two-pm decomposition principle is recunively applied in a hierarchicd rnanner. It also discusses the heuristic parailel aigorithm. a simple extension of the two-part decomposition principle to multi-pan problerns without the hierarchical decomposition pnnciple. Chapter 5 presents the design and implementation procedures of the parallel decomposition solver and shows pteliminary computational results. In each of the tests, the pardiel algorithm converged to within a small tolerance of the optimal solution in a reasonable
Page 1 and 2: A PARALLEL PRIMAL-DUAL DECOMPOSITIO
Page 3 and 4: 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: 1.2 Motivation and Objectives of th
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
Page 63 and 64: 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
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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
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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
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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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