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CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION FOR THE DIRECTED GRAPH - ALGORITHM
1997,17(1):85-90
CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION FOR THE DIRECTED GRAPH - ALGORITHM 1 Zheng Huirao ( ~;I ~ ) Fei Pusheng ( f Department of !'Aathematics, Wuhan University, Wuhan 430072, China.
Fang Yunlan ( 7j.z:: ~)
iM.1.. )
Abstract In this paper we discuss the convergence of the directed graph-algorithm for solving a kind of optimization problems where the objective and subjective functions are all separable, and the parallel implementation process for the directed graph -algorithm is introduced.
Key words separable function, directed graph-algorithm, Jar-metric Princple,state variable, binary directed edge.
1 Introd uction Assuming X = (Xl. :/:'] ..... :I: n
f
t::: R", if f(X) : R"
r--+
R can be formed as follows:
(1.1)
where e denotes a binary opertion(such as +,x, max, min etc.), we call f(X) a spearable function[l]. L.Cooper stated' that most of functions may be transformed to separable forms by increasing assistant variables and constraints[2]. In Practical applications, multistage production-inventory problems, equipment replacement problems, resouce allocation problems and some scheduling problems etc., can all be formulated as the following forms[3]. n
minf(X) = ~ fi(Xi),
(1.2)
i=l
E';=l hj(xj) ~ b h·(x·) J J > - 0, J" = I···n h(X) =
where Ii, hj(l ~ j ~ n) : R r--+ R. Apparaently, the objective and subjective function of the problem(1.2) are all separable. Supposing that, for any b E [0, b], the set {x E R I h j (Xj) = b} 1 Received
Jan.17,1995; revised May 16, 1995. Supported by NNSF of China
86
ACTA MA.1.'HEMATICA SCIENTIA
Vol.17
is non-empty, we have put forward a directed graph-algorithm (In brief we call it DGAlgorithm) for solving the problem(1.2)[4]. In this paper we mainly discuss the convergence and the parallel implementation of the DG-Algorithm for solving the problem(1.2). In the next section.we briefly describe the DG-Algorithm. Section 3 contains the convergence analysis for the algorithm. In Section 4 we introduce its parallel implementation process and finally some concluding remarks are given in Section 5.
2 Directed Graph-Algorithm The DG-Algorithm is discussed in a given semi-field according to the characteristics of the practical problemsl'v. For the problem(1.2), we define the semi-field as {R, /\, +}~ where R = R U{+oo} and R is the real number set, the meet operation /\,called as the modi-addition, denotes the computation of finding the minimum betwee two real numbers in R, and + is ordinary addition. The DG-Algorithm is a discretization method. Here we discuss the discrete case of the variable[2] . Assuming that the discretization stepsize for state variable is D.b, we determine the decision variables xj(l :::; j :::; n) in natural order[2]. In the following, we briefly' describe the DG-Algorithm, where we determine the sets of state-vertices and the values of each binary directed edge(we denote it by bidi-edge in the following) from the first stage [n+l][4], and the symbol]-] denotes the integer part of number, where is less than or equal to the real number.
DG-Algorithm [4] Step 1 Determine the sets of state-vertices. The beginning state-vertex set V(O) = {b} = {v~O)} and the number of its elements k is to = 1. The state-vertex set of the stage k(1 :::; k :::; n) is V(k) = {vi ) , ... , vi~)} = .. (k) (k) (k) { 0, Sb, 2D.b, ... ,b}, here "i < v 2 < ... < "i, . If b] D.b IS an intcger.zj, = [b/D.b] + 1, or else t k = [b/D.b] + 2. We see that t = tk(1 :::; k :::; n) = t l , at last the state-vertex set of the n l n l stage n+ 1 is v(n+l) = + ) is only a mark, having no meaning (Figure 2.1). + ) } , here
{vl
vl
Figure 2.1 Step 2 Determine the values of the bidi-edges.
At the k-th (1 ::; k :::; n) stage, for the edge from the state-vertices v~k-l)(1 ::; i ::; tk-l) to v~k) (1 ::; s ::;tk), if vy-l) 2 v~k), let the non-empty set H = {x I hk ( z ) = v~k-l)~v~k), z E R}, then we define the values of the bidi-edge as (x~k), a~k») 28 28
= (x~k) ,Jl"k(x~k»)) 28
I
28'
where x~k) E 28
(x~;), a~;») == (rLull, +00), where +00 is zero element in the semi-field {R,!\, +} and null suggest that there isn't a real number x such that hk(x) = 'V~k-l) _ v~k).
H, or else we let
87
Fang et al: CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION
No.1
. At the stage n
+ 1,
we let (X1~+1), a~~+l») = (null, 0)(1 ~ i ~ tn), where 0 is identity
element in the same semi-field. Step 3 Relevant to the jar-metric of each stage(We call second value of each bidi -edge as jar-metricl'"), write out all modi-matrices A(i)(i = 1, ... , n + 1).
a(i) 12
a(i) 1t
(i) a 21 a 22
a(i) 2t
(i)
all A(i)
(i)
=
(i) at1
wh ere ~.
= 2 ,... ,n, A(i)·IS
modi-vector, A(n+l) =
(2.1)
vdots (i)
(i) at2
att
° A n d A(l) a t· x t mo dO' i-mat rIX.
(a~~+l) a~~+l)
•••
a~~+l){ is a
= «1) all
(1») a(1) 12 ... a 1t
° IS
a row
column modi-vector.
Step 4 According to jar-metric Principle, compute the modi-product of all A(i)(1 ~ i~n+1):
(2.2) where ~ denotes the modi-multiplication of two modi-matricesld. In the end we can get the optimal discretization solution for the problem( 1.2). From the DG-Algorithm, we can conclude that at each stage, the number of all element in the stage-vertex set can't be too big, less than or equal to t.
3 Convergence Analysis In this section we prove that when some kinds of assumptions are satisfied for I and hj ( 1 :::; j ~ n),we have I(X) --+ c for /lb --+ 0, where X is the optimal discretization solution according to the DG-Algorithm and c is the optimal value of the problem (1.2).That is the following theorem:
Theorem ~.1 In the problem(1.2), suppose that the objective function I satisfies the Lipschitz condition: I I(X 1) - I(X 2) I:::; LII X 1-X2 11 p for all X 1,X2 E Rn,L > 0 is the Lipschitz constant and II . II; denotes the p-norm in R" space, moreover, for each Then for any
E
>
>
0, such that I hj(xj) - hj(xj) 12 0,3/l > 0, when /lb < /l, we have
hj(1 ~ j:::; n),3Lj
I I(X) -
c
t., 1Xj
-.Xj
1for all Xj,Xj
r:< E
where c is the optimal value of the problem(1.2), X discretization solution when the stepsize is Llb.
E R.
(3.1)
= (Xl, X2, .•. , xn)T
is the optimal
Proof Assuming that X* = (xi, xi, ... , x:)T is one of the optimal solutions for the problem(1.2) and I(X*) = c, relevant to X* ,the state-vertices of each stage are Va, respectively. Apparently, in each stage of Figure(2.1) there exists discretization state-vertices V1,. · · ,Vn , such that 1vi - Vi 1< /lb(O :::; i :::; n).And relevant to VA, V1, . . . ,vn ,the value of each decision variable is Xl, X2, ... , Xn , respectively. Then we have (for 1 :::; i :::; n):
vr, ...,v;,
va,
hi(x;)
= vi
- Vi-1'
hi(Xi)
= Vi -
Vi-1'
I Vi --- Vi-1 -
(vi - Vi-1) I~ 2/lb,
88
ACTA MATHEMATICA SCIENTIA
Vol.17
. So
Lilxi -
x;1 :::;
IVi - Vi-l - (v; -
v:_l)1 :::; 2~b.
Let L o == min {Li}'.i == (Xl, X2, .•. , xn)T, then lsz~n
II X"'"
- X*
II 00 ==
m~x
0sasn
I Xi,. .,
-
Xi*
I:::;-L 2~b 0
(3.2)
According to the equivalence of vector norm in R", there exists a constant Co > 0, which has nothing to do with X and X*, such that
. II X -
X*
lip:::; Co II X -
X*
1100
(3.3)
So let fJ == LOE/2Lco and· when ~b < fJ,assuming that X == (Xl,X2, ... ,xn )T is the optimal discretization solution based on the DG-Algorithm, then we conclude from (3.2) and (3.3) that
I f(X)
- f(X*) I == f(X) - f(X*) :::; f(X) - f(X*)
==1 f(X) - f(X*) I:::; L II X - X* lip ob < Lc 0 II X - X* II 00_ < 2LCo~b < 2Lc La La _ -
That is, when
~b
<
s,
2LL oc oe _ 2L o L co -
E
I f(X)
1< E.
- c
From Theorem 3.1, we can obtain Corollary 1 Assuming that X* == (xi, xi, ... , x~)T is one of the optimal solutions for the, problem(1.2), and for each hj(l :::; j :::; n), the condition of Theorem 3.1 is satisfied, then, there exists one discretization feasible solution i == (Xl, X2, ... xn ) in Figure(2.1), such
II i -
X* lip:::; M ~b, where M > 0 is constant. Corollary 2 Assuming that the function f in the problem(1.2) is continuous and for each hj (1 :::; j :::; n), t~e condition of Theorem 3.1 is satisfied, let X is the optimal discretization solution for the stepsize ~b, then, when ~b -+ 0, f(X) -+ c, where c is the optimal value of the problem(1.2). Given the stepsize sequence {~b(q)}, ~b(l) > ~b(2) >,'. ·,and ~b(q) -+ 0 for q ---+ that
+00. Relevantly, according to the DG-Algorithm, we get the optimal discretization solution
} i X(q) =: (-(q) -(q) -(q))T . Th en we h ave teo h £ II OWIng . t h eorem: sequence {uiq) X Xl ,x 2 , ••• , xn Theorem 3.2 If D, the feasible set of the problem(1.2), is a non-empty bounded and closed set, assuming that, in the problem(1.2), the function f is continuous, and for each hj(l :::; j :::; n),the condition of Theorem 3.1 is satisfied,then,for the sequence {X(q)},there
exist convergent subsequences, and for each convergent subsequence {X(qk)}, its limit point
{X*} is an optimal solution of the problem(1.2) . Proof Since D C R" is a non-empty bounded and closed set, then the sequence {X(q)} D f'IS continuous, . . a su bsequence {v
Fang et at: CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION
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89
From Corollary 2, when t1b(q) ~ 0, I(rq/e») ~ c. where c is the optimal value of the " -* -*-* problem(1.2). Therefore, I(X ) =" c. Since XED, X is one optimal solution of the problem(1.2).
4 Parallel Implementation for DG-Algorithm From the DG-Algorithm in Section 2 the operations mainly concentrate on the step2 and step4. Because the modi-multiplication ~ for modi-matrices are associative, the parallel computation of modi-product of the step4 is easy to implement. On the other hand,while discretization, the computations for the step4 are thoroughly ind-ependent on each stage and also convergent to parallel implementation. Assuming that the parallel system has P = 2m processors Pl,P2,··· ,Pq. Firstly we compute the sum of the operations for DG-Algorithm.Using the p-partitioning algorihml'", we partition the directed graph of n+ 1 stages into p parts,each part including the successive k; (1 :S i :S p) stages. of the directed graph, such that, in each part the operations for computing the values of the bidi-edges and the modi-product of the k i modi-matrices are approximate. So the parallel implementation steps for DG-Algorithm are as follows: PDG-Algorithm Stepl Determine the stepsize t1b and the semi-field, evaluate the total operations for DG Algorithm from left to right or from right to left. Step2 Using the D-process of the p-partitioning algorithm repeatedly[6], partition the n + 1 stages into P parts and get all ki(l ~ i ~ p) such that the total operations for each part are approximate. Step3 For each processor Pi(l :::; i :S p), in parallel, compute all x~:) and a~:), where 1 ~ i ~ Ajk j
==
tk-l,l :::; 8 :::; tk
and k from
each part. Here, s; = E~=l kj(l :::; Step4 For k == 1 to m, let
Uj
+ 1 to 8i. then compute the modi-product and each processor records the optimum path of
8i-l
A(8j-l +1) ~ A(Sj-l +2) ~ · · · ~ A(Sj),
i·:::; p) and
== 1 + 2k (j - 1), Vj == i : 2k., 8 ==
Uj
80
=
+ 2k - l
o. -
1,1 ==
8
+1
A u j k• ~ processor PVj ,A1k vj ~ processor PUj' compute all modi-matrices A u j k vJ == AuJok• ~A'kv J and record the optimum path in processor PUj together with processor PVj for aliI:::; j ~ 2m - k in parallel. 0
0
If mod(j, 2) == 1, Au JokVj ~ processor Pu J0' else A U.J k "i ~ processor Pv Jo. At last, the l-order modi-matrix A , k p in processor PI is the optimal discretization value. From the optimum path we get the optimal discretization solution X = (Xl, X2, .. . , xn)T.
5 Several Remarks In this section we give some remarks as follows: (1) Many practical problems are discrete themselves and modi-matrices from the DGAlgorithm ;are often sparse, so can be computed using the algorithm for computing the blocked modi-matrices.
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ACTA MATHEMATICA SCIENTIA
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(2) When,the objective and subjective functions of the optimization problem have other separable forms, we can also compute its approximate solution in a different semi-field,similar to the DG-Algorithm. (3) When using DG-Algorithm, the dec~sion variables may be determined not in natural order. References 1 Mo Cormick G P. Concerting General Nonlinear Progranuning Problems to Separable Nonlinear Programming Problems. George Washington University, Serial T-267,June 1972. 2 Cooper L, Cooper M W. Introduction to Dynamic Progranuning. Pergamon Press,1981. 3 Bellman R E, Drcyfus S E. Applied Dynamic Progrmming. Princeton Unierssity Press,Princeton,1962. 4 Fang Yunlan, Zheng Huirao. A directed Graph-Algoritlun for Solving a kind of N-Dimensioned Optimization Problems.In:Thery and application on Optimization( The Collection of Theses of The second National Conference on Optimization), Xi' an University of Electronice and Teclmology Press(in Chinese),1994.328-332. 5 Qin Yuyuan. Optimum Path Problems In Networks. Hubei Education Press,1992. 6 Zheng Huirao,Fan Rong, Fei Pusheng. A Parallel Algorithm to The Multi-stage Decision Problem of Finite Type. Journal of Wuhan University(in Chinese),1995,1,27-31.
CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION FOR THE DIRECTED GRAPH - ALGORITHM 1 Zheng Huirao ( ~;I ~ ) Fei Pusheng ( f Department of !'Aathematics, Wuhan University, Wuhan 430072, China.
Fang Yunlan ( 7j.z:: ~)
iM.1.. )
Abstract In this paper we discuss the convergence of the directed graph-algorithm for solving a kind of optimization problems where the objective and subjective functions are all separable, and the parallel implementation process for the directed graph -algorithm is introduced.
Key words separable function, directed graph-algorithm, Jar-metric Princple,state variable, binary directed edge.
1 Introd uction Assuming X = (Xl. :/:'] ..... :I: n
f
t::: R", if f(X) : R"
r--+
R can be formed as follows:
(1.1)
where e denotes a binary opertion(such as +,x, max, min etc.), we call f(X) a spearable function[l]. L.Cooper stated' that most of functions may be transformed to separable forms by increasing assistant variables and constraints[2]. In Practical applications, multistage production-inventory problems, equipment replacement problems, resouce allocation problems and some scheduling problems etc., can all be formulated as the following forms[3]. n
minf(X) = ~ fi(Xi),
(1.2)
i=l
E';=l hj(xj) ~ b h·(x·) J J > - 0, J" = I···n h(X) =
where Ii, hj(l ~ j ~ n) : R r--+ R. Apparaently, the objective and subjective function of the problem(1.2) are all separable. Supposing that, for any b E [0, b], the set {x E R I h j (Xj) = b} 1 Received
Jan.17,1995; revised May 16, 1995. Supported by NNSF of China
86
ACTA MA.1.'HEMATICA SCIENTIA
Vol.17
is non-empty, we have put forward a directed graph-algorithm (In brief we call it DGAlgorithm) for solving the problem(1.2)[4]. In this paper we mainly discuss the convergence and the parallel implementation of the DG-Algorithm for solving the problem(1.2). In the next section.we briefly describe the DG-Algorithm. Section 3 contains the convergence analysis for the algorithm. In Section 4 we introduce its parallel implementation process and finally some concluding remarks are given in Section 5.
2 Directed Graph-Algorithm The DG-Algorithm is discussed in a given semi-field according to the characteristics of the practical problemsl'v. For the problem(1.2), we define the semi-field as {R, /\, +}~ where R = R U{+oo} and R is the real number set, the meet operation /\,called as the modi-addition, denotes the computation of finding the minimum betwee two real numbers in R, and + is ordinary addition. The DG-Algorithm is a discretization method. Here we discuss the discrete case of the variable[2] . Assuming that the discretization stepsize for state variable is D.b, we determine the decision variables xj(l :::; j :::; n) in natural order[2]. In the following, we briefly' describe the DG-Algorithm, where we determine the sets of state-vertices and the values of each binary directed edge(we denote it by bidi-edge in the following) from the first stage [n+l][4], and the symbol]-] denotes the integer part of number, where is less than or equal to the real number.
DG-Algorithm [4] Step 1 Determine the sets of state-vertices. The beginning state-vertex set V(O) = {b} = {v~O)} and the number of its elements k is to = 1. The state-vertex set of the stage k(1 :::; k :::; n) is V(k) = {vi ) , ... , vi~)} = .. (k) (k) (k) { 0, Sb, 2D.b, ... ,b}, here "i < v 2 < ... < "i, . If b] D.b IS an intcger.zj, = [b/D.b] + 1, or else t k = [b/D.b] + 2. We see that t = tk(1 :::; k :::; n) = t l , at last the state-vertex set of the n l n l stage n+ 1 is v(n+l) = + ) is only a mark, having no meaning (Figure 2.1). + ) } , here
{vl
vl
Figure 2.1 Step 2 Determine the values of the bidi-edges.
At the k-th (1 ::; k :::; n) stage, for the edge from the state-vertices v~k-l)(1 ::; i ::; tk-l) to v~k) (1 ::; s ::;tk), if vy-l) 2 v~k), let the non-empty set H = {x I hk ( z ) = v~k-l)~v~k), z E R}, then we define the values of the bidi-edge as (x~k), a~k») 28 28
= (x~k) ,Jl"k(x~k»)) 28
I
28'
where x~k) E 28
(x~;), a~;») == (rLull, +00), where +00 is zero element in the semi-field {R,!\, +} and null suggest that there isn't a real number x such that hk(x) = 'V~k-l) _ v~k).
H, or else we let
87
Fang et al: CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION
No.1
. At the stage n
+ 1,
we let (X1~+1), a~~+l») = (null, 0)(1 ~ i ~ tn), where 0 is identity
element in the same semi-field. Step 3 Relevant to the jar-metric of each stage(We call second value of each bidi -edge as jar-metricl'"), write out all modi-matrices A(i)(i = 1, ... , n + 1).
a(i) 12
a(i) 1t
(i) a 21 a 22
a(i) 2t
(i)
all A(i)
(i)
=
(i) at1
wh ere ~.
= 2 ,... ,n, A(i)·IS
modi-vector, A(n+l) =
(2.1)
vdots (i)
(i) at2
att
° A n d A(l) a t· x t mo dO' i-mat rIX.
(a~~+l) a~~+l)
•••
a~~+l){ is a
= «1) all
(1») a(1) 12 ... a 1t
° IS
a row
column modi-vector.
Step 4 According to jar-metric Principle, compute the modi-product of all A(i)(1 ~ i~n+1):
(2.2) where ~ denotes the modi-multiplication of two modi-matricesld. In the end we can get the optimal discretization solution for the problem( 1.2). From the DG-Algorithm, we can conclude that at each stage, the number of all element in the stage-vertex set can't be too big, less than or equal to t.
3 Convergence Analysis In this section we prove that when some kinds of assumptions are satisfied for I and hj ( 1 :::; j ~ n),we have I(X) --+ c for /lb --+ 0, where X is the optimal discretization solution according to the DG-Algorithm and c is the optimal value of the problem (1.2).That is the following theorem:
Theorem ~.1 In the problem(1.2), suppose that the objective function I satisfies the Lipschitz condition: I I(X 1) - I(X 2) I:::; LII X 1-X2 11 p for all X 1,X2 E Rn,L > 0 is the Lipschitz constant and II . II; denotes the p-norm in R" space, moreover, for each Then for any
E
>
>
0, such that I hj(xj) - hj(xj) 12 0,3/l > 0, when /lb < /l, we have
hj(1 ~ j:::; n),3Lj
I I(X) -
c
t., 1Xj
-.Xj
1for all Xj,Xj
r:< E
where c is the optimal value of the problem(1.2), X discretization solution when the stepsize is Llb.
E R.
(3.1)
= (Xl, X2, .•. , xn)T
is the optimal
Proof Assuming that X* = (xi, xi, ... , x:)T is one of the optimal solutions for the problem(1.2) and I(X*) = c, relevant to X* ,the state-vertices of each stage are Va, respectively. Apparently, in each stage of Figure(2.1) there exists discretization state-vertices V1,. · · ,Vn , such that 1vi - Vi 1< /lb(O :::; i :::; n).And relevant to VA, V1, . . . ,vn ,the value of each decision variable is Xl, X2, ... , Xn , respectively. Then we have (for 1 :::; i :::; n):
vr, ...,v;,
va,
hi(x;)
= vi
- Vi-1'
hi(Xi)
= Vi -
Vi-1'
I Vi --- Vi-1 -
(vi - Vi-1) I~ 2/lb,
88
ACTA MATHEMATICA SCIENTIA
Vol.17
. So
Lilxi -
x;1 :::;
IVi - Vi-l - (v; -
v:_l)1 :::; 2~b.
Let L o == min {Li}'.i == (Xl, X2, .•. , xn)T, then lsz~n
II X"'"
- X*
II 00 ==
m~x
0sasn
I Xi,. .,
-
Xi*
I:::;-L 2~b 0
(3.2)
According to the equivalence of vector norm in R", there exists a constant Co > 0, which has nothing to do with X and X*, such that
. II X -
X*
lip:::; Co II X -
X*
1100
(3.3)
So let fJ == LOE/2Lco and· when ~b < fJ,assuming that X == (Xl,X2, ... ,xn )T is the optimal discretization solution based on the DG-Algorithm, then we conclude from (3.2) and (3.3) that
I f(X)
- f(X*) I == f(X) - f(X*) :::; f(X) - f(X*)
==1 f(X) - f(X*) I:::; L II X - X* lip ob < Lc 0 II X - X* II 00_ < 2LCo~b < 2Lc La La _ -
That is, when
~b
<
s,
2LL oc oe _ 2L o L co -
E
I f(X)
1< E.
- c
From Theorem 3.1, we can obtain Corollary 1 Assuming that X* == (xi, xi, ... , x~)T is one of the optimal solutions for the, problem(1.2), and for each hj(l :::; j :::; n), the condition of Theorem 3.1 is satisfied, then, there exists one discretization feasible solution i == (Xl, X2, ... xn ) in Figure(2.1), such
II i -
X* lip:::; M ~b, where M > 0 is constant. Corollary 2 Assuming that the function f in the problem(1.2) is continuous and for each hj (1 :::; j :::; n), t~e condition of Theorem 3.1 is satisfied, let X is the optimal discretization solution for the stepsize ~b, then, when ~b -+ 0, f(X) -+ c, where c is the optimal value of the problem(1.2). Given the stepsize sequence {~b(q)}, ~b(l) > ~b(2) >,'. ·,and ~b(q) -+ 0 for q ---+ that
+00. Relevantly, according to the DG-Algorithm, we get the optimal discretization solution
} i X(q) =: (-(q) -(q) -(q))T . Th en we h ave teo h £ II OWIng . t h eorem: sequence {uiq) X Xl ,x 2 , ••• , xn Theorem 3.2 If D, the feasible set of the problem(1.2), is a non-empty bounded and closed set, assuming that, in the problem(1.2), the function f is continuous, and for each hj(l :::; j :::; n),the condition of Theorem 3.1 is satisfied,then,for the sequence {X(q)},there
exist convergent subsequences, and for each convergent subsequence {X(qk)}, its limit point
{X*} is an optimal solution of the problem(1.2) . Proof Since D C R" is a non-empty bounded and closed set, then the sequence {X(q)} D f'IS continuous, . . a su bsequence {v
Fang et at: CONVERGENCE ANALYSIS AND PARALLEL IMPLEMENTATION
No.1
89
From Corollary 2, when t1b(q) ~ 0, I(rq/e») ~ c. where c is the optimal value of the " -* -*-* problem(1.2). Therefore, I(X ) =" c. Since XED, X is one optimal solution of the problem(1.2).
4 Parallel Implementation for DG-Algorithm From the DG-Algorithm in Section 2 the operations mainly concentrate on the step2 and step4. Because the modi-multiplication ~ for modi-matrices are associative, the parallel computation of modi-product of the step4 is easy to implement. On the other hand,while discretization, the computations for the step4 are thoroughly ind-ependent on each stage and also convergent to parallel implementation. Assuming that the parallel system has P = 2m processors Pl,P2,··· ,Pq. Firstly we compute the sum of the operations for DG-Algorithm.Using the p-partitioning algorihml'", we partition the directed graph of n+ 1 stages into p parts,each part including the successive k; (1 :S i :S p) stages. of the directed graph, such that, in each part the operations for computing the values of the bidi-edges and the modi-product of the k i modi-matrices are approximate. So the parallel implementation steps for DG-Algorithm are as follows: PDG-Algorithm Stepl Determine the stepsize t1b and the semi-field, evaluate the total operations for DG Algorithm from left to right or from right to left. Step2 Using the D-process of the p-partitioning algorithm repeatedly[6], partition the n + 1 stages into P parts and get all ki(l ~ i ~ p) such that the total operations for each part are approximate. Step3 For each processor Pi(l :::; i :S p), in parallel, compute all x~:) and a~:), where 1 ~ i ~ Ajk j
==
tk-l,l :::; 8 :::; tk
and k from
each part. Here, s; = E~=l kj(l :::; Step4 For k == 1 to m, let
Uj
+ 1 to 8i. then compute the modi-product and each processor records the optimum path of
8i-l
A(8j-l +1) ~ A(Sj-l +2) ~ · · · ~ A(Sj),
i·:::; p) and
== 1 + 2k (j - 1), Vj == i : 2k., 8 ==
Uj
80
=
+ 2k - l
o. -
1,1 ==
8
+1
A u j k• ~ processor PVj ,A1k vj ~ processor PUj' compute all modi-matrices A u j k vJ == AuJok• ~A'kv J and record the optimum path in processor PUj together with processor PVj for aliI:::; j ~ 2m - k in parallel. 0
0
If mod(j, 2) == 1, Au JokVj ~ processor Pu J0' else A U.J k "i ~ processor Pv Jo. At last, the l-order modi-matrix A , k p in processor PI is the optimal discretization value. From the optimum path we get the optimal discretization solution X = (Xl, X2, .. . , xn)T.
5 Several Remarks In this section we give some remarks as follows: (1) Many practical problems are discrete themselves and modi-matrices from the DGAlgorithm ;are often sparse, so can be computed using the algorithm for computing the blocked modi-matrices.
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(2) When,the objective and subjective functions of the optimization problem have other separable forms, we can also compute its approximate solution in a different semi-field,similar to the DG-Algorithm. (3) When using DG-Algorithm, the dec~sion variables may be determined not in natural order. References 1 Mo Cormick G P. Concerting General Nonlinear Progranuning Problems to Separable Nonlinear Programming Problems. George Washington University, Serial T-267,June 1972. 2 Cooper L, Cooper M W. Introduction to Dynamic Progranuning. Pergamon Press,1981. 3 Bellman R E, Drcyfus S E. Applied Dynamic Progrmming. Princeton Unierssity Press,Princeton,1962. 4 Fang Yunlan, Zheng Huirao. A directed Graph-Algoritlun for Solving a kind of N-Dimensioned Optimization Problems.In:Thery and application on Optimization( The Collection of Theses of The second National Conference on Optimization), Xi' an University of Electronice and Teclmology Press(in Chinese),1994.328-332. 5 Qin Yuyuan. Optimum Path Problems In Networks. Hubei Education Press,1992. 6 Zheng Huirao,Fan Rong, Fei Pusheng. A Parallel Algorithm to The Multi-stage Decision Problem of Finite Type. Journal of Wuhan University(in Chinese),1995,1,27-31.