Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol26.Issue1.2000:
Operations Research Letters 26 (2000) 33–41
www.elsevier.com/locate/orms
Simplicial with truncated Dantzig–Wolfe decomposition
for nonlinear multicommodity network
ow problems
with side constraints (
Siriphong Lawphongpanich ∗
Department of Operations Research, Naval Postgraduate School, 1411 Cunningham Road, Monterey, CA 93943, USA
Received 1 June 1998; received in revised form 1 July 1999
Abstract
The simplicial decomposition (SD) subproblem for a nonlinear multicommodity network
ow problem is simply its linear
approximation. Instead of solving the subproblem optimally, this paper demonstrates that performing one iteration of Dantzig–
c 2000 Elsevier Science
Wolfe decomposition is generally sucient for SD to eciently converge to an optimal solution.
B.V. All rights reserved.
Keywords: Multicommodity network; Trac assignment problem; Decomposition technique
1. Introduction
The nonlinear multicommodity network
ow problem with side constraints can be stated as follows:
!
K
X
NMNFP-SC : min f
x(k)
x
k=1
s:t: Bx(k) = b(k);
K
X
∀k = 1; : : : ; K;
Sx(k)6u;
k=1
x(k)¿0;
∀k = 1; : : : ; K;
( This research was partially supported by the U.S. Army’s
Oce of the Deputy Chief of Sta for Personnel and the Naval
Postgraduate School Institutionally Funded Research Program.
∗ Fax: +1-831-656-2595.
E-mail address: slawphon@nps.navy.mil (S. Lawphongpanich)
where B is the node-arc incidence matrix for the
underlying network, b(k) is the supply–demand vector for commodity k; S and u are a matrix and a
vector, respectively, forming the side constraints,
x(k) is the
ow vector for commodity k, and f(x)
is a pseudo-convex cost function. For the remainder, P
X represents the aggregate
ow vector, i.e.,
K
X = k=1 x(k).
One important instance of NMNFP-SC and the main
motivation of this work is the capacitated trac assignment problem (see, e.g., [13,18]) where S is the
identity matrix and u is a vector of arc capacities.
However, there are other applications such as those
in, e.g., [2,22,23].
Simplicial decomposition (SD) as applied to
NMNFP-SC can be stated as follows:
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 9 9 ) 0 0 0 5 9 - 0
34
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Simplicial Decomposition
Step 0: Let X 1 be a feasible aggregate
ow vector
and set = 1.
Step 1: Solve
K
X
SD-SP( ): Y = arg min
y
to a set of shortest path problems. On the other hand,
the resulting master problem, i.e.,
(
!
X
X
+1
r
= arg min f
r Y :
r SY r 6u;
t
3f(X ) y(k)
r=1
X
k=1
s:t: By(k) = b(k); ∀k = 1; : : : ; K;
K
X
Sy(k)6u;
k=1
y(k)¿0; ∀k = 1; : : : ; K;
where y(k) and Y are de ned in a manner similar to
x(k) and X , respectively.
Step 2: If 3f(X )t (Y − X )¿0, stop and X is
an optimal solution.
Step 3: Solve
SD-MP( ):
+1 = arg min
(
f
X
!
r Y r :
r=1
X
r = 1
r=1
)
and r ¿0; ∀r = 1; : : : ; :
Then, set X +1 =
Go to Step 1.
P
r=1
)
r = 1; r ¿0; ∀r = 1; : : : ;
r=1
is larger and more complex. Below, it is demonstrated
that the master problem can be kept as simple as the
one in Step 3 while maintaining the shortest path structure of the subproblem.
To solve SD-SP( ), it is natural to consider decomposition techniques (see, e.g., [17]), especially for
large networks. One such technique is the Dantzig–
Wolfe (DW) decomposition (see [5]) which decomposes SD-SP( ) into the following two problems, a
master (DW-MP) and a subproblem (DW-SP). The
master problem can be stated as follows:
DW-MP :
min
s:t:
Q
X
(3f(X )t P q )q
q=1
Q
X
q SP q 6u;
q=1
r +1 Y r and set = + 1.
In Step 1, problem SD-SP( ) is a linear multicommodity network
ow problem with side constraints.
(See, e.g., [2].) Because of its size, solving SD-SP( )
optimally, or nearly so, is time consuming. To avoid
doing so, many (see, e.g., [13,23,18]) have dualized
or penalized the side constraints in NMNFP-SC. The
resulting dual or penalty problem is a nonlinear multicommodity network
ow problem without the side
constraints for which there are several ecient algorithms (see, e.g., [7]). On the other hand, Gon et al.
[10] and Hearn and Lawphongpanich [11] used variations of the cutting plane technique (see, e.g., [3]) to
solve a Lagrangian dual of NMNFP-SC instead.
Using a di erent strategy for decomposing
NMNFP-SC, Rutenberg [24] (see also [16]), Marin
[22], and Wu and Ventura [28] delete the side constraints from SD-SP( ) and add them to the master
problem. Doing so reduces the resulting subproblem
r=1
Q
X
q = 1;
q=1
q ¿0; ∀q = 1; : : : ; Q;
where P q ; ∀q = 1; : : : ; Q, are extreme points of the
following set:
(
K
X
p(k); Bp(k) = b(k); and p(k)¿0;
F = P: P =
k=1
)
∀k = 1; : : : ; K :
In words, F is the set of aggregate
ow vectors that
satisfy the
ow balance constraint of every commodity. These
ow vectors, however, may not be feasible to NMNFP-SC because they may violate the side
constraints. In DW decomposition, only a subset of
the extreme points of F is included in the DW-MP
35
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
initially. The remaining extreme points are generated
as needed by solving the following subproblem:
Step 1: Solve
n
DW-SP : min
p
K
X
DW-MP( ; n): = arg min
(∇f(X ) − S t )t p(k)
where is the dual vector corresponding to the side
constraints (or the rst set of constraints) in DW-MP.
Observe that DW-SP can be decomposed into K
separate shortest path problems.
To establish a benchmark for his own algorithms,
Stefek [25] implemented a restricted version of SD
(see, e.g., [12]) and solved SD-SP( ) by DW decomposition to near optimality. This typically requires a
large number of DW iterations, i.e., solving a large
number of DW-MP and DW-SP problems.
Instead of obtaining a near optimal solution to
SD-SP( ), the next section demonstrates that DW
decomposition, when ‘nested’ in simplicial decomposition, can be truncated after one iteration. The idea
of ‘nesting’, or recursively applying DW decomposition, is described earlier in [14,9] for staircase or
multi-stage linear programs. (For additional enhancements, see [27] and references cited therein.) The term
‘nesting’ is used here because SD is essentially DW
decomposition generalized to nonlinear problems.
Finally, Section 3 summarizes the numerical results
from an implementation of the algorithm in an algebraic modeling system, GAMS [4].
2. Simplicial with truncated Dantzig–Wolfe
decomposition
Below is a version of SD nested with truncated
DW decomposition. In this version, superscripts distinguish di erent vectors and subscripts denote di erent components of a vector. Among the superscripts,
and n index extreme points in the SD and DW master
problems, respectively.
Simplicial with truncated Dantzig–Wolfe (STDW)
decomposition
Step 0: LetPX 1 be a feasible aggregate
ow vecK
tor such that k=1 Sx1 (k)6(1 − )u for some small
¿ 0. Set P 1 = X 1 ; Y 0 = X 1 ; = 1 and n = 1.
(3f(X )t P q )q
q=1
k=1
s:t: Bp(k) = b(k); ∀k = 1; : : : ; K;
p(k)¿0; ∀k = 1; : : : ; K;
n
X
s:t:
n
X
q SP q 6u;
q=1
n
X
q = 1;
q=1
q ¿0; ∀q = 1; : : : ; n;
where q is the weight for the qth extreme point and
= (1 ; 2 ; : : : ; n )t .
Let n and
n be optimal dual variables corresponding to the capacity
Pnand convexity constraints, respectively. Set Z n = q=1 qn P q .
Step 2: If 3f(X )t (Z n − X ) ¡ 0, then set Y = Z n
and go to Step 5. Otherwise, go to step 3.
Step 3: For each k = 1; : : : ; K, solve
DW-SP(n):
pn+1 (k) = arg min
(3f(X ) − S t n )t p
p
s:t:
Bp = b(k);
p¿0:
PK
Set P n+1 = k=1 pn+1 (k). If (3f(X )−S t n )t P n+1 −
n ¿0, stop and X is optimal. Otherwise, go to Step 4.
Step 4:P
Solve DW-MP( ; n + 1) (see Step 1). Then,
n+1
set Y = q=1 qn+1 P q and n = n + 1. Go to Step 5.
Step 5: Solve
SD-MP( ):
+1
= arg min
(
f
X
r=0
r Y
r
!
:
X
r = 1
r=0
)
and r ¿0; ∀r = 0; : : : ;
Then, set X +1 =
to Step 1.
P
r=0
r +1 Y r and set = + 1. Go
Step 0 requires a solution, X 1 , that is both feasible
and interior with respect to the side constraints. If
36
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
none is available, apply DW decomposition to the following problem until the arti cial vector w becomes
zero:
y
K
X
s:t:
By(k) = b(k);
min
3f(X̂ )t y(k) + Met w
k=1
K
X
∀k = 1; : : : ; K;
Sy(k) − w6u;
k=1
y(k)¿0;
∀k = 1; : : : ; K;
w¿0;
where M is a suitably large constant, e is a (column) vector of ones, and the aggregate
ow vector
X̂ is arbitrary. Alternately, 3f(X̂ ) can be replaced
by a zero vector and the above problem becomes
the Phase-I problem of the 2-Phase method for linear programs. However, our numerical study suggests
that using 3f(X̂ ) can lead to a faster convergence.
In any case, when DW decomposition terminates, it
must yield a set of extreme points, i.e., {P 1 ; : : : ; P n}
where P
n is a positive integer, and a vector such that
n
X 1 = q=1 q P q is feasible to NMNFP-SC. At this
point, set Y 0 = X 1 ; = 1 and n = n and go to Step 1.
In Step 1, problem DW-MP( ; n) as stated includes all of the previously generated extreme points
of F. (This requirement may be relaxed and it is
a subject for a subsequent article.) Then, Step 2
checks whether a new feasible solution, Z n , leads
to an improvement or generates a descent direction.
If so, solving the SD-MP( ) in Step 5 would yield
a new aggregate
ow vector, X +1 , with a smaller
objective function. When Z n does not satisfy the
condition in Step 2, Step 3 generates a new extreme
point, P n+1 , and adds it to the next master problem,
DW-MP( ; n + 1), solved in Step 4. The theorem
below shows that the next feasible point, Y , derived from the solution of DW-MP( ; n + 1) must
yield a descent direction under a mild condition.
Theorem 2.1. Assume that X is not optimal
and DW-MP( ; n) has a unique optimal solution.
If 3f(X )t (Z n − X )¿0; then 3f(X )t (Y −
X ) ¡ 0; where Y is derived from a solution to
DW-MP( ; n + 1).
Proof. Since each Y r ; r = 0; : : : ; ( − 1), is in
the convex hull of {P 1 ; : : : ; P n } and X is a convex combination of Y 0 ; : : : ; Y ( −1) ; X must be in
the convex hull as well. From the condition that
3f(X )t (Z n − X )¿0, the unique solution,
n , to
Pn
n q
DW-MP( ; n) must be such that X = q=1 q P . To
verify, 3f(X )t (Z n − X )¿0 implies that
3f(X )t X 63f(X )t Z n =
n
X
(3f(X )t P n )qn :
q=1
Since n solves DW-MP( ; n) uniquely, the inequality
in the above expression must hold at equality and X =
P
n
n q
q=1 q P .
To obtain a contradiction, assume that 3f(X )t (Y
− X )¿0. This implies that a solution n+1 such that
qn+1
=
(
qn
if q = 1; : : : ; n;
0
if q = n + 1
is optimal to DW-MP( ; n+1). Moreover, ( n ;
n ), the
optimal dual variables for DW-MP( ; n), must be optimal to the dual of DW-MP( ; n+1) also. Given these
dual values, the reduced cost for q is (3f(X ) −
S t n )t P q −
n , which must be nonnegative for all
q = 1; : : : ; n + 1. In particular,
(3f(X ) − S t n )t P q −
n = 0;
n
X
∀q: qn+1 ¿ 0;
[(3f(X ) − S t n )t P q −
n ]qn+1 = 0;
q=1
(3f(X ) − S t n )t X −
n = 0;
where the last equality follows from the construction
of n+1 and the fact that qn sums to one. Since the
reduced cost for n+1 is nonnegative, the following
must hold:
(3f(X ) − S t n )t P n+1 −
n
¿(3f(X ) − S n )t X −
n = 0;
(3f(X ) − S t n )t (P n+1 − X )¿0;
(3f(X ) − S t n )t (P − X )¿0;
∀P ∈ F;
37
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
where the last inequality follows from the fact that
pn+1 (k) solves the kth shortest path problem in
DW-SP(n). However, this shows that X satis es the
optimality condition for NMNFP-SC (see, e.g., [22])
which is a contradiction.
Because solving DW-SP(n) in Step 3 and
DW-MP( ; n + 1) in Step 4 constitutes one iteration
of DW decomposition, the above theorem demonstrates that a descent direction for SD-MP( ) can be
obtained by truncating the decomposition after one
iteration.
When 3f(X )t (Z n − X )¿0, an alternate optimal
solution to DW-MP( ; n) may exist. In such a case, Z n
may not equal X and the wrong set of dual variables
(i.e., those associated with Z n instead of X ) may be
transferred to DW-SP(n). Thus, 3f(X )t (Y − X )
may not be negative as in Theorem 2.1. To ensure the
same result when 3f(X )t (Z n − X )¿0 and Z n 6=
X , one method is to let ( n ;
n ) be a solution to the
following problem:
max
( ;
)
s:t:
ut +
q t
(3f(X ) − S t n )t (P − X )¿0;
t
q
∀q = 1; : : : ; n;
60;
3. Implementation
The STDW algorithm in Section 2 was implemented
in GAMS, see [4]. In our implementation, the aggregate
ow vector P q is disaggregated into K vectors,
one for each demand k (see, [19]), and DW-MP( ; n)
is replaced with the following.
DDW-MP( ; n):
Without the last constraint, the above problem is the
dual of DW-MP( ; n) in Step 1. On the other hand,
the last constraint is derived from the complementary
slackness condition associated with the rst constraint
in DW-MP( ; n) and the optimal primal solution X .
This ensures that the dual solution ( n ;
n ) and X
form an optimal primal–dual pair.
When the algorithm terminates in Step 3, two conditions are satis ed. The rst condition is in Step 2
and veri es that Z n does not produce a descent direction. The second is in Step 3 and it ensures that no
descent direction is possible. The justi cation of these
two criteria is given in the following theorem.
Theorem 2.2. If 3f(X )t (Z n − X )¿0 and (3f(X )
− S t n )t P n+1 −
n ¿0; then X is optimal.
Proof. In a manner similar to the proof of Theorem
2.1, it can be demonstrated that these two conditions
K
n X
X
(3f(X )t pq (k))q (k)
q=1 k=1
s:t:
∀i: [SX ]i ¡ ui :
∀P ∈ S:
Steps 1– 4 produce a descent direction for the
master problem in Step 5 or they verify that
the current solution is optimal. This observation
forms a basis for the convergence of STDW,
which can be established in the same manner
as that of simplicial decomposition. (See, e.g.,
[15,26].)
n = arg min
[SP ] +
63f(X ) P ;
t = 0
lead to the following optimality condition:
K
n X
X
q (k)Spq (k)6u;
q=1 k=1
n
X
q (k) = 1;
∀k = 1; : : : ; K;
q=1
q ¿0;
∀q = 1; : : : ; n:
Although not implemented here, Y r can also be
disaggregated or represented as (yr (1); : : : ; yr (K)) in
SD-MP( ).
so, some components of the vecPK PDoing
tor k=1 r=1 r (k)yr (k) may violate the side constraints even though no component of Y r does. So,
the side constraints must be added to the SD-MP( )
to ensure that they are not violated. This results in a
larger and more complex master problem.
Also, the condition in Step 2 is replaced by the
following:
3f(X )t Z n 63f(X )t X
for some 0 ¡ ¡ 1:
In our implementation, is set to 0.999 to insure the
di erence between 3f(X )t Z n and 3f(X )t X is not
due to numerical inaccuracy.
38
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Table 1
Information on ve nonlinear multicommodity network
ow problems
Problem
# Nodes
# Arcs
# OD pairs
References
Nine-node
Sioux falls
Hull
NDO22
NDO148
9
24
501
14
61
18
76
798
22
148
4
528
142
23
122
Hearn and Ribera [13]
LeBlanc et al. [21] and Abdulaal and LeBlanc [1]
Florian et al. [6]
Gon et al. [10]
Gon et al. [10]
Table 2
Computational results for ve nonlinear multicommodity network
ow problems using STDW
Network
Capacity
Obtaining an initial
feasible solution
Main iterations
Objective
function
DW-MP DW-SP
SD-MP DW-MP DW-SP
Lower
bound
Relative
gap
% Arcs with
ow ¿0:9∗ cap
Nine node
Original
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
7
5
5
3
7
5
5
3
2
1
1
0
5
3
2
1
3
2
1
1
2291.6747
1912.5211
1873.3279
1829.5694
2291.6747
1912.5210
1873.3276
1829.5694
0.0000
0.0000
0.0000
0.0000
66.67
71.43
57.14
42.86
Sioux falls
2:0∗ Orig.
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
3
3
3
2
3
3
3
2
29
13
28
45
45
23
42
63
16
10
14
18
43.2764
42.5357
42.3789
42.3178
43.2743
42.5319
42.3768
42.3136
0.0001
0.0001
0.0001
0.0001
36.84
76.32
55.26
17.11
Hull
1:512∗ Orig.
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
3
6
5
4
3
6
5
4
24
3
5
10
41
7
11
18
17
4
6
8
34809.0456
35070.0681
34970.2294
34930.6648
34806.7409
35069.8717
34968.4551
34927.2595
0.0001
0.0000
0.0001
0.0001
1.80
81.52
63.24
18.48
NDO22a
NDO148a
Original
Original
2
2
2
2
7
58
14
75
7
16
103.4121
151.9368
103.4120 0.0001
151.9269 0.0001
9.09
0.00
a Lower
bounds for communication networks are from [10] and they are rounded to four digits.
Finally, our implementation of STDW begins by
using DW decomposition to nd an initial feasible solution and terminates when the relative gap,
(f(X ) − lower bound)=lower bound, is less than
0.0001. Whenever Step 3 is executed, the lower
bound of the optimal objective function value can be
obtained via the following result.
Theorem 3.1. Let X ∗ denote an optimal solution to
NMNFP-SC. Then;
f(X ∗ ) ¿ f(X ) + 3f(X )t (P n+1 − X )
+ (u − SP n+1 )t n :
Proof. The result follows from the inequalities below.
(
f(X ) ¿ f(X ) + min 3f(X )t
∗
y
×
K
X
y(k) − X
k=1
!
:
K
X
Sy(k)6u;
k=1
By(k) = b(k); y(k)¿0; ∀k
)
¿ f(X ) + L( ∗ )
¿ f(X ) + L( n )
= f(X ) + 3f(X )t (P n+1 − X )
+ (u − SP n+1 )t n ;
Table 3
Previous results for ve nonlinear multicommodity network
ow problems
Prob.
Method used
Objective
value
Num. shortest
path cal.
Remarks
Nine-node
Hearn and Ribera [13]
CTAPa
Aug. Lagrangian
2292.64
250
Larsson and Patriksson
[18]
CTAP
Aug. Lagrangian
2291.31
Not speci ed
Lawphongpanich and
Hearn [20]
UTAPb
Restricted SD
1455.19
6
The authors solved 25 augmented Lagrangian problems by
performing 10 Frank–Wolfe iterations for each one. In the nal
solution, two arcs exceed their capacities by 0:01 and one by
0:02.
The authors solved 25 augmented Lagrangian problems via
disaggregate simplicial decomposition. Their heuristic algorithm
did not nd a feasible solution and the nal solution has two arcs
exceeding their capacities by 0:01 and one by 0:02.
The authors used di erent parameters for the travel cost
functions.
UTAP
Modi ed
Frank–Wolfe
Disaggregate SD
42.33
50
42.31
5
CTAP
Aug. Lagrangian
and heuristic
alg.
43.37
Not speci ed
UTAP
UTAP
Restricted SD
Disaggregate SD
34,808.18
21,516.64
27
4
Communication
networks
Gon et al. [10]
NDO22
103.41202
14
Gon et al. [10]
NDO148
Analytic center
cutting plane
Analytic center
cutting plane
151.92687
16
Sioux falls
Fukushima [8]
Larsson and Patriksson
[19]
Larsson and Patriksson
[18]
Hull
Hearn et al. [12]
Larsson and Patriksson
[19]
a CTAP
b UTAP
UTAP
The authors solved two augmented Lagrangain problems via
disaggregate simplicial decomposition and used a heuristic
algorithm to obtain a feasible solution.
This problem uses the same network, but other data are di erent.
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Reference
The number in the objective value column is the best known
Lagrangain dual value, i.e., lower bound.
The number in the objective value column is the best known
Lagrangain dual value, i.e., lower bound.
= Capacitated Trac Assignment Problem.
= Uncapacitated Trac Assignment Problem.
39
40
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
where L( ) is the following Lagrangian dual function:
!
(
K
X
y(k) − X
L( ) = min 3f(X )t
y
k=1
+ u−
K
X
!t
Sy(k)
k=1
y(k)¿0; ∀k
: By(k) = b(k);
)
and ∗ is an optimal dual solution, i.e., ∗ =
arg max{L( ): 60}.
The rst three inequalities above hold, respectively,
because f(X ) is pseudo-convex, the Weak Duality
Theorem applies, and n does not solve the dual problem. The last equality follows from the fact that P n+1 =
P
K
n+1
(k) solves the DW-SP(n).
k=1 p
Five problems from the literature were selected for
testing. Three are trac assignment problems and the
remaining two are problems in network communication. Their statistics are given in Table 1.
For all problems,
P the travel cost function is separable, i.e., f(x) = (i; j)∈
fij (xij ), where
is the set
of arcs in the network, and each arc cost, fij (xij ), is
generally a convex function of its capacity. For Sioux
Falls and Hull, these capacities are too small, for they
make the resulting capacitated trac assignment problem (CTAP) infeasible. To construct feasible CTAP,
the original capacities are multiplied by a factor ¿ 1,
when necessary. Alternately, the system optimal solution (see [7]) can serve as arc capacities. However,
using system optimal
ows as arc capacities tends to
reduce the feasible region of CTAP to a single point
and STDW would terminate as soon as it nds an initial feasible solution. To generate more meaningful
problems, the capacities are set to ×(system optimal
solution) where ¿ 1.
Table 2 summarizes the results for the ve
test problems using STDW. The table primarily
lists the number of times each problem (SD-MP,
DW-MP, and DW-SP) must be solved to achieve
0.0001 relative gap or better. For large networks,
a good indicator of an algorithm’s eciency is the
number of shortest path calculations or the number of DW-SPs solved. (See, e.g., [12].) Com-
paring these numbers in Table 2 (particularly
those in the main iterations, i.e., the iterations after an initial feasible solution has been obtained)
with those reported in the literature (see Table
3) demonstrates that STDW is competitive with
the existing algorithms solving the same or similar problems. When compared with penalty or
dual-based algorithms, STDW o ers an added advantage, in that it always produces a feasible solution when terminated prior to reaching an optimal
solution.
Acknowledgements
This research was completed while the author was
spending his sabbatical leave at the Center for Applied Optimization and Department of Industrial and
Systems Engineering at the University of Florida. He
is grateful for the support provided by these two organizations. He would like to also thank Prof. Hearn for
providing the data for this research, Prof. Ramana for
suggesting the use of system optimal solutions, and
Prof. Morton for Ref. [27].
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Pennsylvania State University, 1994.
www.elsevier.com/locate/orms
Simplicial with truncated Dantzig–Wolfe decomposition
for nonlinear multicommodity network
ow problems
with side constraints (
Siriphong Lawphongpanich ∗
Department of Operations Research, Naval Postgraduate School, 1411 Cunningham Road, Monterey, CA 93943, USA
Received 1 June 1998; received in revised form 1 July 1999
Abstract
The simplicial decomposition (SD) subproblem for a nonlinear multicommodity network
ow problem is simply its linear
approximation. Instead of solving the subproblem optimally, this paper demonstrates that performing one iteration of Dantzig–
c 2000 Elsevier Science
Wolfe decomposition is generally sucient for SD to eciently converge to an optimal solution.
B.V. All rights reserved.
Keywords: Multicommodity network; Trac assignment problem; Decomposition technique
1. Introduction
The nonlinear multicommodity network
ow problem with side constraints can be stated as follows:
!
K
X
NMNFP-SC : min f
x(k)
x
k=1
s:t: Bx(k) = b(k);
K
X
∀k = 1; : : : ; K;
Sx(k)6u;
k=1
x(k)¿0;
∀k = 1; : : : ; K;
( This research was partially supported by the U.S. Army’s
Oce of the Deputy Chief of Sta for Personnel and the Naval
Postgraduate School Institutionally Funded Research Program.
∗ Fax: +1-831-656-2595.
E-mail address: slawphon@nps.navy.mil (S. Lawphongpanich)
where B is the node-arc incidence matrix for the
underlying network, b(k) is the supply–demand vector for commodity k; S and u are a matrix and a
vector, respectively, forming the side constraints,
x(k) is the
ow vector for commodity k, and f(x)
is a pseudo-convex cost function. For the remainder, P
X represents the aggregate
ow vector, i.e.,
K
X = k=1 x(k).
One important instance of NMNFP-SC and the main
motivation of this work is the capacitated trac assignment problem (see, e.g., [13,18]) where S is the
identity matrix and u is a vector of arc capacities.
However, there are other applications such as those
in, e.g., [2,22,23].
Simplicial decomposition (SD) as applied to
NMNFP-SC can be stated as follows:
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 9 9 ) 0 0 0 5 9 - 0
34
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Simplicial Decomposition
Step 0: Let X 1 be a feasible aggregate
ow vector
and set = 1.
Step 1: Solve
K
X
SD-SP( ): Y = arg min
y
to a set of shortest path problems. On the other hand,
the resulting master problem, i.e.,
(
!
X
X
+1
r
= arg min f
r Y :
r SY r 6u;
t
3f(X ) y(k)
r=1
X
k=1
s:t: By(k) = b(k); ∀k = 1; : : : ; K;
K
X
Sy(k)6u;
k=1
y(k)¿0; ∀k = 1; : : : ; K;
where y(k) and Y are de ned in a manner similar to
x(k) and X , respectively.
Step 2: If 3f(X )t (Y − X )¿0, stop and X is
an optimal solution.
Step 3: Solve
SD-MP( ):
+1 = arg min
(
f
X
!
r Y r :
r=1
X
r = 1
r=1
)
and r ¿0; ∀r = 1; : : : ; :
Then, set X +1 =
Go to Step 1.
P
r=1
)
r = 1; r ¿0; ∀r = 1; : : : ;
r=1
is larger and more complex. Below, it is demonstrated
that the master problem can be kept as simple as the
one in Step 3 while maintaining the shortest path structure of the subproblem.
To solve SD-SP( ), it is natural to consider decomposition techniques (see, e.g., [17]), especially for
large networks. One such technique is the Dantzig–
Wolfe (DW) decomposition (see [5]) which decomposes SD-SP( ) into the following two problems, a
master (DW-MP) and a subproblem (DW-SP). The
master problem can be stated as follows:
DW-MP :
min
s:t:
Q
X
(3f(X )t P q )q
q=1
Q
X
q SP q 6u;
q=1
r +1 Y r and set = + 1.
In Step 1, problem SD-SP( ) is a linear multicommodity network
ow problem with side constraints.
(See, e.g., [2].) Because of its size, solving SD-SP( )
optimally, or nearly so, is time consuming. To avoid
doing so, many (see, e.g., [13,23,18]) have dualized
or penalized the side constraints in NMNFP-SC. The
resulting dual or penalty problem is a nonlinear multicommodity network
ow problem without the side
constraints for which there are several ecient algorithms (see, e.g., [7]). On the other hand, Gon et al.
[10] and Hearn and Lawphongpanich [11] used variations of the cutting plane technique (see, e.g., [3]) to
solve a Lagrangian dual of NMNFP-SC instead.
Using a di erent strategy for decomposing
NMNFP-SC, Rutenberg [24] (see also [16]), Marin
[22], and Wu and Ventura [28] delete the side constraints from SD-SP( ) and add them to the master
problem. Doing so reduces the resulting subproblem
r=1
Q
X
q = 1;
q=1
q ¿0; ∀q = 1; : : : ; Q;
where P q ; ∀q = 1; : : : ; Q, are extreme points of the
following set:
(
K
X
p(k); Bp(k) = b(k); and p(k)¿0;
F = P: P =
k=1
)
∀k = 1; : : : ; K :
In words, F is the set of aggregate
ow vectors that
satisfy the
ow balance constraint of every commodity. These
ow vectors, however, may not be feasible to NMNFP-SC because they may violate the side
constraints. In DW decomposition, only a subset of
the extreme points of F is included in the DW-MP
35
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
initially. The remaining extreme points are generated
as needed by solving the following subproblem:
Step 1: Solve
n
DW-SP : min
p
K
X
DW-MP( ; n): = arg min
(∇f(X ) − S t )t p(k)
where is the dual vector corresponding to the side
constraints (or the rst set of constraints) in DW-MP.
Observe that DW-SP can be decomposed into K
separate shortest path problems.
To establish a benchmark for his own algorithms,
Stefek [25] implemented a restricted version of SD
(see, e.g., [12]) and solved SD-SP( ) by DW decomposition to near optimality. This typically requires a
large number of DW iterations, i.e., solving a large
number of DW-MP and DW-SP problems.
Instead of obtaining a near optimal solution to
SD-SP( ), the next section demonstrates that DW
decomposition, when ‘nested’ in simplicial decomposition, can be truncated after one iteration. The idea
of ‘nesting’, or recursively applying DW decomposition, is described earlier in [14,9] for staircase or
multi-stage linear programs. (For additional enhancements, see [27] and references cited therein.) The term
‘nesting’ is used here because SD is essentially DW
decomposition generalized to nonlinear problems.
Finally, Section 3 summarizes the numerical results
from an implementation of the algorithm in an algebraic modeling system, GAMS [4].
2. Simplicial with truncated Dantzig–Wolfe
decomposition
Below is a version of SD nested with truncated
DW decomposition. In this version, superscripts distinguish di erent vectors and subscripts denote di erent components of a vector. Among the superscripts,
and n index extreme points in the SD and DW master
problems, respectively.
Simplicial with truncated Dantzig–Wolfe (STDW)
decomposition
Step 0: LetPX 1 be a feasible aggregate
ow vecK
tor such that k=1 Sx1 (k)6(1 − )u for some small
¿ 0. Set P 1 = X 1 ; Y 0 = X 1 ; = 1 and n = 1.
(3f(X )t P q )q
q=1
k=1
s:t: Bp(k) = b(k); ∀k = 1; : : : ; K;
p(k)¿0; ∀k = 1; : : : ; K;
n
X
s:t:
n
X
q SP q 6u;
q=1
n
X
q = 1;
q=1
q ¿0; ∀q = 1; : : : ; n;
where q is the weight for the qth extreme point and
= (1 ; 2 ; : : : ; n )t .
Let n and
n be optimal dual variables corresponding to the capacity
Pnand convexity constraints, respectively. Set Z n = q=1 qn P q .
Step 2: If 3f(X )t (Z n − X ) ¡ 0, then set Y = Z n
and go to Step 5. Otherwise, go to step 3.
Step 3: For each k = 1; : : : ; K, solve
DW-SP(n):
pn+1 (k) = arg min
(3f(X ) − S t n )t p
p
s:t:
Bp = b(k);
p¿0:
PK
Set P n+1 = k=1 pn+1 (k). If (3f(X )−S t n )t P n+1 −
n ¿0, stop and X is optimal. Otherwise, go to Step 4.
Step 4:P
Solve DW-MP( ; n + 1) (see Step 1). Then,
n+1
set Y = q=1 qn+1 P q and n = n + 1. Go to Step 5.
Step 5: Solve
SD-MP( ):
+1
= arg min
(
f
X
r=0
r Y
r
!
:
X
r = 1
r=0
)
and r ¿0; ∀r = 0; : : : ;
Then, set X +1 =
to Step 1.
P
r=0
r +1 Y r and set = + 1. Go
Step 0 requires a solution, X 1 , that is both feasible
and interior with respect to the side constraints. If
36
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
none is available, apply DW decomposition to the following problem until the arti cial vector w becomes
zero:
y
K
X
s:t:
By(k) = b(k);
min
3f(X̂ )t y(k) + Met w
k=1
K
X
∀k = 1; : : : ; K;
Sy(k) − w6u;
k=1
y(k)¿0;
∀k = 1; : : : ; K;
w¿0;
where M is a suitably large constant, e is a (column) vector of ones, and the aggregate
ow vector
X̂ is arbitrary. Alternately, 3f(X̂ ) can be replaced
by a zero vector and the above problem becomes
the Phase-I problem of the 2-Phase method for linear programs. However, our numerical study suggests
that using 3f(X̂ ) can lead to a faster convergence.
In any case, when DW decomposition terminates, it
must yield a set of extreme points, i.e., {P 1 ; : : : ; P n}
where P
n is a positive integer, and a vector such that
n
X 1 = q=1 q P q is feasible to NMNFP-SC. At this
point, set Y 0 = X 1 ; = 1 and n = n and go to Step 1.
In Step 1, problem DW-MP( ; n) as stated includes all of the previously generated extreme points
of F. (This requirement may be relaxed and it is
a subject for a subsequent article.) Then, Step 2
checks whether a new feasible solution, Z n , leads
to an improvement or generates a descent direction.
If so, solving the SD-MP( ) in Step 5 would yield
a new aggregate
ow vector, X +1 , with a smaller
objective function. When Z n does not satisfy the
condition in Step 2, Step 3 generates a new extreme
point, P n+1 , and adds it to the next master problem,
DW-MP( ; n + 1), solved in Step 4. The theorem
below shows that the next feasible point, Y , derived from the solution of DW-MP( ; n + 1) must
yield a descent direction under a mild condition.
Theorem 2.1. Assume that X is not optimal
and DW-MP( ; n) has a unique optimal solution.
If 3f(X )t (Z n − X )¿0; then 3f(X )t (Y −
X ) ¡ 0; where Y is derived from a solution to
DW-MP( ; n + 1).
Proof. Since each Y r ; r = 0; : : : ; ( − 1), is in
the convex hull of {P 1 ; : : : ; P n } and X is a convex combination of Y 0 ; : : : ; Y ( −1) ; X must be in
the convex hull as well. From the condition that
3f(X )t (Z n − X )¿0, the unique solution,
n , to
Pn
n q
DW-MP( ; n) must be such that X = q=1 q P . To
verify, 3f(X )t (Z n − X )¿0 implies that
3f(X )t X 63f(X )t Z n =
n
X
(3f(X )t P n )qn :
q=1
Since n solves DW-MP( ; n) uniquely, the inequality
in the above expression must hold at equality and X =
P
n
n q
q=1 q P .
To obtain a contradiction, assume that 3f(X )t (Y
− X )¿0. This implies that a solution n+1 such that
qn+1
=
(
qn
if q = 1; : : : ; n;
0
if q = n + 1
is optimal to DW-MP( ; n+1). Moreover, ( n ;
n ), the
optimal dual variables for DW-MP( ; n), must be optimal to the dual of DW-MP( ; n+1) also. Given these
dual values, the reduced cost for q is (3f(X ) −
S t n )t P q −
n , which must be nonnegative for all
q = 1; : : : ; n + 1. In particular,
(3f(X ) − S t n )t P q −
n = 0;
n
X
∀q: qn+1 ¿ 0;
[(3f(X ) − S t n )t P q −
n ]qn+1 = 0;
q=1
(3f(X ) − S t n )t X −
n = 0;
where the last equality follows from the construction
of n+1 and the fact that qn sums to one. Since the
reduced cost for n+1 is nonnegative, the following
must hold:
(3f(X ) − S t n )t P n+1 −
n
¿(3f(X ) − S n )t X −
n = 0;
(3f(X ) − S t n )t (P n+1 − X )¿0;
(3f(X ) − S t n )t (P − X )¿0;
∀P ∈ F;
37
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
where the last inequality follows from the fact that
pn+1 (k) solves the kth shortest path problem in
DW-SP(n). However, this shows that X satis es the
optimality condition for NMNFP-SC (see, e.g., [22])
which is a contradiction.
Because solving DW-SP(n) in Step 3 and
DW-MP( ; n + 1) in Step 4 constitutes one iteration
of DW decomposition, the above theorem demonstrates that a descent direction for SD-MP( ) can be
obtained by truncating the decomposition after one
iteration.
When 3f(X )t (Z n − X )¿0, an alternate optimal
solution to DW-MP( ; n) may exist. In such a case, Z n
may not equal X and the wrong set of dual variables
(i.e., those associated with Z n instead of X ) may be
transferred to DW-SP(n). Thus, 3f(X )t (Y − X )
may not be negative as in Theorem 2.1. To ensure the
same result when 3f(X )t (Z n − X )¿0 and Z n 6=
X , one method is to let ( n ;
n ) be a solution to the
following problem:
max
( ;
)
s:t:
ut +
q t
(3f(X ) − S t n )t (P − X )¿0;
t
q
∀q = 1; : : : ; n;
60;
3. Implementation
The STDW algorithm in Section 2 was implemented
in GAMS, see [4]. In our implementation, the aggregate
ow vector P q is disaggregated into K vectors,
one for each demand k (see, [19]), and DW-MP( ; n)
is replaced with the following.
DDW-MP( ; n):
Without the last constraint, the above problem is the
dual of DW-MP( ; n) in Step 1. On the other hand,
the last constraint is derived from the complementary
slackness condition associated with the rst constraint
in DW-MP( ; n) and the optimal primal solution X .
This ensures that the dual solution ( n ;
n ) and X
form an optimal primal–dual pair.
When the algorithm terminates in Step 3, two conditions are satis ed. The rst condition is in Step 2
and veri es that Z n does not produce a descent direction. The second is in Step 3 and it ensures that no
descent direction is possible. The justi cation of these
two criteria is given in the following theorem.
Theorem 2.2. If 3f(X )t (Z n − X )¿0 and (3f(X )
− S t n )t P n+1 −
n ¿0; then X is optimal.
Proof. In a manner similar to the proof of Theorem
2.1, it can be demonstrated that these two conditions
K
n X
X
(3f(X )t pq (k))q (k)
q=1 k=1
s:t:
∀i: [SX ]i ¡ ui :
∀P ∈ S:
Steps 1– 4 produce a descent direction for the
master problem in Step 5 or they verify that
the current solution is optimal. This observation
forms a basis for the convergence of STDW,
which can be established in the same manner
as that of simplicial decomposition. (See, e.g.,
[15,26].)
n = arg min
[SP ] +
63f(X ) P ;
t = 0
lead to the following optimality condition:
K
n X
X
q (k)Spq (k)6u;
q=1 k=1
n
X
q (k) = 1;
∀k = 1; : : : ; K;
q=1
q ¿0;
∀q = 1; : : : ; n:
Although not implemented here, Y r can also be
disaggregated or represented as (yr (1); : : : ; yr (K)) in
SD-MP( ).
so, some components of the vecPK PDoing
tor k=1 r=1 r (k)yr (k) may violate the side constraints even though no component of Y r does. So,
the side constraints must be added to the SD-MP( )
to ensure that they are not violated. This results in a
larger and more complex master problem.
Also, the condition in Step 2 is replaced by the
following:
3f(X )t Z n 63f(X )t X
for some 0 ¡ ¡ 1:
In our implementation, is set to 0.999 to insure the
di erence between 3f(X )t Z n and 3f(X )t X is not
due to numerical inaccuracy.
38
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Table 1
Information on ve nonlinear multicommodity network
ow problems
Problem
# Nodes
# Arcs
# OD pairs
References
Nine-node
Sioux falls
Hull
NDO22
NDO148
9
24
501
14
61
18
76
798
22
148
4
528
142
23
122
Hearn and Ribera [13]
LeBlanc et al. [21] and Abdulaal and LeBlanc [1]
Florian et al. [6]
Gon et al. [10]
Gon et al. [10]
Table 2
Computational results for ve nonlinear multicommodity network
ow problems using STDW
Network
Capacity
Obtaining an initial
feasible solution
Main iterations
Objective
function
DW-MP DW-SP
SD-MP DW-MP DW-SP
Lower
bound
Relative
gap
% Arcs with
ow ¿0:9∗ cap
Nine node
Original
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
7
5
5
3
7
5
5
3
2
1
1
0
5
3
2
1
3
2
1
1
2291.6747
1912.5211
1873.3279
1829.5694
2291.6747
1912.5210
1873.3276
1829.5694
0.0000
0.0000
0.0000
0.0000
66.67
71.43
57.14
42.86
Sioux falls
2:0∗ Orig.
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
3
3
3
2
3
3
3
2
29
13
28
45
45
23
42
63
16
10
14
18
43.2764
42.5357
42.3789
42.3178
43.2743
42.5319
42.3768
42.3136
0.0001
0.0001
0.0001
0.0001
36.84
76.32
55.26
17.11
Hull
1:512∗ Orig.
1:05∗ sys-opt
1:10∗ sys-opt
1:20∗ sys-opt
3
6
5
4
3
6
5
4
24
3
5
10
41
7
11
18
17
4
6
8
34809.0456
35070.0681
34970.2294
34930.6648
34806.7409
35069.8717
34968.4551
34927.2595
0.0001
0.0000
0.0001
0.0001
1.80
81.52
63.24
18.48
NDO22a
NDO148a
Original
Original
2
2
2
2
7
58
14
75
7
16
103.4121
151.9368
103.4120 0.0001
151.9269 0.0001
9.09
0.00
a Lower
bounds for communication networks are from [10] and they are rounded to four digits.
Finally, our implementation of STDW begins by
using DW decomposition to nd an initial feasible solution and terminates when the relative gap,
(f(X ) − lower bound)=lower bound, is less than
0.0001. Whenever Step 3 is executed, the lower
bound of the optimal objective function value can be
obtained via the following result.
Theorem 3.1. Let X ∗ denote an optimal solution to
NMNFP-SC. Then;
f(X ∗ ) ¿ f(X ) + 3f(X )t (P n+1 − X )
+ (u − SP n+1 )t n :
Proof. The result follows from the inequalities below.
(
f(X ) ¿ f(X ) + min 3f(X )t
∗
y
×
K
X
y(k) − X
k=1
!
:
K
X
Sy(k)6u;
k=1
By(k) = b(k); y(k)¿0; ∀k
)
¿ f(X ) + L( ∗ )
¿ f(X ) + L( n )
= f(X ) + 3f(X )t (P n+1 − X )
+ (u − SP n+1 )t n ;
Table 3
Previous results for ve nonlinear multicommodity network
ow problems
Prob.
Method used
Objective
value
Num. shortest
path cal.
Remarks
Nine-node
Hearn and Ribera [13]
CTAPa
Aug. Lagrangian
2292.64
250
Larsson and Patriksson
[18]
CTAP
Aug. Lagrangian
2291.31
Not speci ed
Lawphongpanich and
Hearn [20]
UTAPb
Restricted SD
1455.19
6
The authors solved 25 augmented Lagrangian problems by
performing 10 Frank–Wolfe iterations for each one. In the nal
solution, two arcs exceed their capacities by 0:01 and one by
0:02.
The authors solved 25 augmented Lagrangian problems via
disaggregate simplicial decomposition. Their heuristic algorithm
did not nd a feasible solution and the nal solution has two arcs
exceeding their capacities by 0:01 and one by 0:02.
The authors used di erent parameters for the travel cost
functions.
UTAP
Modi ed
Frank–Wolfe
Disaggregate SD
42.33
50
42.31
5
CTAP
Aug. Lagrangian
and heuristic
alg.
43.37
Not speci ed
UTAP
UTAP
Restricted SD
Disaggregate SD
34,808.18
21,516.64
27
4
Communication
networks
Gon et al. [10]
NDO22
103.41202
14
Gon et al. [10]
NDO148
Analytic center
cutting plane
Analytic center
cutting plane
151.92687
16
Sioux falls
Fukushima [8]
Larsson and Patriksson
[19]
Larsson and Patriksson
[18]
Hull
Hearn et al. [12]
Larsson and Patriksson
[19]
a CTAP
b UTAP
UTAP
The authors solved two augmented Lagrangain problems via
disaggregate simplicial decomposition and used a heuristic
algorithm to obtain a feasible solution.
This problem uses the same network, but other data are di erent.
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
Reference
The number in the objective value column is the best known
Lagrangain dual value, i.e., lower bound.
The number in the objective value column is the best known
Lagrangain dual value, i.e., lower bound.
= Capacitated Trac Assignment Problem.
= Uncapacitated Trac Assignment Problem.
39
40
S. Lawphongpanich / Operations Research Letters 26 (2000) 33–41
where L( ) is the following Lagrangian dual function:
!
(
K
X
y(k) − X
L( ) = min 3f(X )t
y
k=1
+ u−
K
X
!t
Sy(k)
k=1
y(k)¿0; ∀k
: By(k) = b(k);
)
and ∗ is an optimal dual solution, i.e., ∗ =
arg max{L( ): 60}.
The rst three inequalities above hold, respectively,
because f(X ) is pseudo-convex, the Weak Duality
Theorem applies, and n does not solve the dual problem. The last equality follows from the fact that P n+1 =
P
K
n+1
(k) solves the DW-SP(n).
k=1 p
Five problems from the literature were selected for
testing. Three are trac assignment problems and the
remaining two are problems in network communication. Their statistics are given in Table 1.
For all problems,
P the travel cost function is separable, i.e., f(x) = (i; j)∈
fij (xij ), where
is the set
of arcs in the network, and each arc cost, fij (xij ), is
generally a convex function of its capacity. For Sioux
Falls and Hull, these capacities are too small, for they
make the resulting capacitated trac assignment problem (CTAP) infeasible. To construct feasible CTAP,
the original capacities are multiplied by a factor ¿ 1,
when necessary. Alternately, the system optimal solution (see [7]) can serve as arc capacities. However,
using system optimal
ows as arc capacities tends to
reduce the feasible region of CTAP to a single point
and STDW would terminate as soon as it nds an initial feasible solution. To generate more meaningful
problems, the capacities are set to ×(system optimal
solution) where ¿ 1.
Table 2 summarizes the results for the ve
test problems using STDW. The table primarily
lists the number of times each problem (SD-MP,
DW-MP, and DW-SP) must be solved to achieve
0.0001 relative gap or better. For large networks,
a good indicator of an algorithm’s eciency is the
number of shortest path calculations or the number of DW-SPs solved. (See, e.g., [12].) Com-
paring these numbers in Table 2 (particularly
those in the main iterations, i.e., the iterations after an initial feasible solution has been obtained)
with those reported in the literature (see Table
3) demonstrates that STDW is competitive with
the existing algorithms solving the same or similar problems. When compared with penalty or
dual-based algorithms, STDW o ers an added advantage, in that it always produces a feasible solution when terminated prior to reaching an optimal
solution.
Acknowledgements
This research was completed while the author was
spending his sabbatical leave at the Center for Applied Optimization and Department of Industrial and
Systems Engineering at the University of Florida. He
is grateful for the support provided by these two organizations. He would like to also thank Prof. Hearn for
providing the data for this research, Prof. Ramana for
suggesting the use of system optimal solutions, and
Prof. Morton for Ref. [27].
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