Journal ArticleDOI

# A comparison of routing sets for robust network design

01 Jun 2014-Optimization Letters (Springer Berlin Heidelberg)-Vol. 8, Iss: 5, pp 1619-1635

TL;DR: This paper compares the optimal capacity allocation costs for six routing sets: affine routing, volume routing and its two simplifications, the routing based on an unrestricted 2-cover of the uncertainty set, and the routingbased on a cover delimited by a hyperplane.

AbstractDesigning a network able to route a set of non-simultaneous demand vectors is an important problem arising in telecommunications. In this paper, we compare the optimal capacity allocation costs for six routing sets: affine routing, volume routing and its two simplifications, the routing based on an unrestricted 2-cover of the uncertainty set, and the routing based on a cover delimited by a hyperplane.

Topics: Static routing (78%), Policy-based routing (77%), Dynamic Source Routing (74%), Multipath routing (72%)

### 1 Introduction

• Given a graph and a set of point-to-point commodities with known demand values, the deterministic network design problem aims at installing enough capacity on the arcs of the graph so that the resulting network is able to route all commodities.
• The introduction of the uncertainty set leads to a robust optimization problem.
• Hence, the routing decisions can be seen as the recourse.
• In Section 3, the authors try to understand how good is the cost of the optimal capacity allocation provided by each of the routing sets, and they compare these costs among the different routing sets.

### 2.1 Problem formulation

• Then, the authors introduce the robust network design problem.
• Notice that in real applications, these costs are usually non-negative.
• In fact, the problem is very difficult to solve given that only deciding whether a given capacity allocation vector x supports D is coNP-complete for general polytopes D, see Chekuri et al. [14] and Gupta et al. [18].
• Frangioni et al. [16] and Poss and Raack [26] show under very strong assumptions on D that the optimal capacity allocations provided by dynamic routings are equivalent to the ones provided by static routings and affine routings, respectively, which are polynomially solvable when D has a compact formulation.

### 2.2.1 Static routing

• This framework considers a restriction on the second stage recourse known as static routing (also called oblivious routing).
• Hence, the resulting optimization problem is polynomially solvable.
• In the following, the authors review alternative routing sets F ′ that are less restrictive than static routings while not being as flexible as dynamic routings.

### 2.2.2 Covers of the uncertainty set delimited by a hyperplane

• Ben-Ameur [3] introduces the idea of covering the uncertainty set by two (or more) subsets using hyperplanes and proposes to use a routing template for each subset.
• The definition above implies that both routing templates y1 and y2 must be able to route demand vectors that lie in the hyperplane {d, αd = β} without exceeding the capacity.
• He further works on the framework in Ben-Ameur and Zotkiewicz [7].

### 2.2.3 Arbitrary covers of the uncertainty set

• Scutellà [27] introduces the idea of using conjointly two routing templates.
• She mentions that the complexity of RND(F2) is unknown.
• The framework described by F2 has been independently proposed for general robust programs by Bertsimas and Caramanis [11] (see also Bertsimas et al. [12]) where the authors propose to cover the uncertainty sets with k subsets and devise independent sets of recourse variables for each of these subsets.

### 2.2.4 Volume routings

• More recently, Ben-Ameur and Zotkiewicz [6] introduce a framework that shares the demand between two routing templates, according to thresholds hk for each k ∈ K.
• They prove that RND(FV) is an NP-hard optimization problem.
• Hence, they introduce simpler frameworks described below.

### 2.2.5 Affine routings

• Ben-Tal et al. [8] introduce Affine Adjustable Robust Counterparts restricting the recourse to be an affine function of the uncertainties.
• This framework has been compared theoretically and numerically to static and dynamic routings by Poss and Raack [26].
• In particular, the authors show that a compact formulation can be described for RND(Faff) as long as D has a compact description, generalizing the result obtained for static routing already.
• The authors point out that a major difference between Faff and the routing described in Section 2.2.1-2.2.4 is that the formers are build up using routing templates, so that it is implicitly assumed that flow conservation constraints (2) and non-negativity constraints (3) are satisfied.
• In opposition, routings in Faff are build up using ordinary vectors so that that satisfaction of (2) and (3) must be stated explicitly.

### 2.3 Contributions of this paper

• The objective of this paper is to compare opt(F ′) among the routing sets recalled in previous sections.
• The polytope introduced by Bertsimas and Sim [13], used for robust network design problems in [6, 23, 24, 20, 26], satisfies the assumption of (b) when the number of deviations allowed is integer.

### 3 Optimal costs

• The objective of this section is to compare the cost of the optimal capacity allocations obtained for RND(F ′) using different routing sets F ′.
• In Section 3.1, the authors describe the methodology used herein to obtain the desired relations.

### 3.1 Methodology

• Proving this inclusion is a very strong result, which holds only for closely related routing sets.
• Because it is not always possible to compare directly the routing sets themselves, the second approach is based on comparing the sets of all capacity allocations that support D when considering a specific routing set.
• The two approaches are formalized in the result below.
• Follows immediately from the definition of RND(F ′), also known as 1.
• Follows from the fact that opt(F ′) is the cost of the optimal solution of (11), also known as 2.

### 3.2 Routings that cover D

• Then, since the inequalities in (6) are not strict, the authors see easily that D̃1 and D̃1 are closed.
• It follows from Proposition 1 that the costs of optimal capacity allocations are always equal.
• The complexity of RND(F2) follows directly from the sufficiency condition of Theorem 1 and the fact that Ben-Ameur [3] proves RND(F2|) to be NP-hard.

### 3.3 Volume and affine routings

• In this section the authors compare volume and affine routings.
• The advantage of decreasing the flow sent on some arcs when the demand for a commodity rises allows to better combine different commodities within the available capacity.
• The problem must contain more than one commodity because dim(D) = 1 and D is orthogonal to the k-th axis.
• (15) Property (15) follows directly from the fact that any routing in FVS is a linear function.
• Thus, let us define Theorem 3 states that whenever D satisfies Assumption 1, one should not try to use the complex set of routings FV, since opt(FV) will never beat opt(FVS).

### 4 Non-comparable routings

• The authors compare opt(F2), opt(FV) and opt(Faff) for general uncertainty sets.
• To devise examples showing that Faff may yield more expensive capacity allocations than F2 and FV, the authors shall use the following result.
• One can easily extend these examples to larger graphs for which each commodity k ∈ K has at least two different paths from its source s(k) to its sink t(k).
• Edge labels from Figure 2(b) and Figure 2(c) represent optimal capacity allocations with dynamic and static routing, respectively.
• A routing f ∈ F that satisfies the capacity from Figure 3(b) is depicted on Figure 3(d) and Figure 3(e), for d1 and d2, respectively.

### 5 Concluding remarks

• This paper studies the optimal capacity allocation cost provided by robust network design models restricted to use specific routing sets.
• Affine routing, volume routing and its two simplifications, and the routings based on covers of the demand uncertainty set, also known as These routing sets are.
• The authors show that the routing set based on an arbitrary cover of the uncertainty is equivalent to the routing set that uses a separation hyperplane.
• An important characteristic of these routing sets is the complexity of the resulting network design problem.
• The general volume routings and the routing sets based on covers of the uncertainty set lead to NP-hard optimization problems.

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A comparison of routing sets for robust network design
Michael Poss
Abstract
Designing a network able to route a set of non-simultaneous demand vectors is an important
problem arising in telecommunications. The problem can be seen a two-stage robust program
where the recourse function consists in choosing the routing for each demand vector. Allowing the
routing to change arbitrarily as the demand varies yields a very diﬃcult optimization problem
so that diﬀerent subsets of admissible routings have been discussed in the literature. In this
paper, we compare theoretically the optimal capacity allocation costs for six of these routing
sets: aﬃne routing, volume routing and its two simpliﬁcations, the routing based on an arbitrary
2-cover of the uncertainty set, and the routing based on a cover delimited by a hyperplane. We
show that the two routing sets based on covers of the uncertainty set yield the same optimal
costs. We show then that the two simpliﬁed volume routings are special cases of aﬃne routings.
Finally, assuming that the uncertainty set is the one studied by Bertsimas and Sim (2004), we
show that the optimal cost provided by volume routing is not less than the costs provided by
the simpliﬁed volume routings.
Keywords: Robust optimization; Network design; Routing set; Routing template; Aﬃne rout-
ing.
1 Introduction
Given a graph and a set of point-to-point commodities with known demand values, the deterministic
network design problem aims at installing enough capacity on the arcs of the graph so that the
resulting network is able to route all commodities. In practice it is however very diﬃcult to know
with precision the exact values of the demands at the time the design decisions are taken. In the
best case, we can estimate a set that contains most likely values for the demand. The introduction
of the uncertainty set leads to a robust optimization problem. In this context, a solution is said
to be feasible for the problem if it is feasible for all demand vectors that belong to the estimated
uncertainty set D, see Soyster [28] and Ben-Tal and Nemirovski [9, 10], among others. This rigid
framework is computationally easy but it does not allow the model to react against the uncertainty.
To address this drawback, Ben-Tal et al. [8] introduce two-stage robust optimization models that
allows to adjust a subset of the problem variables only after observing the actual realization of the
data. This adjusting procedure is often called recourse. This two-stage approach applies naturally
to network design since ﬁrst stage capacity design decisions are usually made in the long term while
the routing decisions depend on the realization of the demand. Hence, the routing decisions can be
seen as the recourse. Free recourse is called dynamic routing in the context of robust network design
problems. It has been shown by Chekuri et al. [14] and Gupta et al. [18] that the robust network
design with dynamic routing is intractable. Already deciding whether or not a ﬁxed capacity design
allows for a dynamic routing of demands in a given polytope is co-N P-complete (on directed graphs).
It is known already that two-stage robust programming with arbitrary recourse is computation-
ally intractable [8]. For this reason, Ben-Tal et al. [8] limit the recourse to aﬃne functions of the
uncertainties which makes the problem tractable. Further works by Chen and Zhang [15] and Goh
and Sim [17] suggest to extend the second stage to piece-wise linear functions of the uncertainties. In
fact, considering special types of recourses had been used already in the context of network design.
Ben-Ameur and Kerivin [4, 5] introduce the concept of static routing: after ﬁxing the design, the
routing of a commodity is allowed to change but only linearly with the variation of the commodity.
Static routing can also be seen as a single stage robust program where the set of routings paths to-
gether with the percental splitting among the paths are chosen at the same time the design decisions
Department of Computer Science, Facult´e des Sciences, Universit´e Libre de Bruxelles, Brussels, Belgium,
mposs@ulb.ac.be
1

are made. The resulting set of paths and percental splitting is often called a routing template, which
is used by all demand vectors in the uncertainty set. The use of static routing makes the robust
network design problem tractable but it yields more expensive capacity allocations than the problem
with dynamic routing. Static routing has been used by various authors since its introduction by
Ben-Ameur and Kerivin, including Altin et al. [1], Koster et al. [19], Ord´o˜nez and Zhao [22].
Several authors tried to introduce routing schemes that are more ﬂexible than static routing
while still being computationally easier than dynamic routing. Ben-Ameur [3] covers the demand
uncertainty set by two (or more) subsets using separating hyperplanes and uses speciﬁc routings
templates for each subset. The resulting optimization problem is N P-hard when no assumptions is
made on the hyperplanes. Scutella [27] generalizes this idea to arbitrary covers of the uncertainty
set. She allows a set of routing templates to be used conjointly so that each demand vector can be
routed by at least one of the routing templates. She also introduces a procedure that works in two
steps. First, an optimal capacity allocation with static routing is computed. Then, she allows allows
to reroute part of the demand vectors according to a second routing template. Ben-Ameur and
Zotkiewicz [6] introduce volume routing, a framework that shares the demand between two routing
templates, according to thresholds. They prove that the resulting optimization problem is N P-
hard and introduce two simpliﬁcations. Finally, applying the aﬃne recourse from Ben-Tal et al. [8]
to robust network design problems, Ouorou and Vial [24] introduce the concept of aﬃne routing.
Recently, Poss and Raack [26, 25] study the properties of aﬃne routing, and compare the later to the
static and dynamic routings, both theoretically and empirically. They conclude that aﬃne routing
tends to yield very good approximations of dynamic routing while being computationally tractable.
In this paper, we compare theoretically the optimal capacity allocation costs provided by the aﬃne
routings from Ouorou and Vial [24], the volume routings from Ben-Ameur and Zotkiewicz [6], and
the routings based on covers of the uncertainty set in two subsets (Ben-Ameur [3] and Scutella [27]).
In the next section, we introduce the robust network design problem and deﬁne a routing set. We
model the robust network design problem with the explicit dependency on the routing set and
formalize each of the routing frameworks studied herein. Our main results are stated in Section 2.3.
In Section 3, we try to understand how good is the cost of the optimal capacity allocation provided
by each of the routing sets, and we compare these costs among the diﬀerent routing sets. We start
by comparing the costs obtained with routings based on covers of the uncertainty set in Section 3.2.
Then, in Section 3.3, we turn to aﬃne and volume routings. In Section 4, we present examples
showing that it is not possible, in general, to compare some of these costs. Finally, we conclude the
paper in Section 5.
2 Robust network design
2.1 Problem formulation
The problem is deﬁned below for a directed graph G = (V, A) and a set of commodities K. We
formalize ﬁrst the concept of a routing. Then, we introduce the robust network design problem.
Each commodity k K has a source s(k) V , a destination t(k) V , and a demand value d
k
0.
A multi-commodity ﬂow is a vector f R
|A|×|K|
+
that satisﬁes the ﬂow conservation constraints at
each node of the network:
X
aδ
+
(v)
f
k
a
X
aδ
(v)
f
k
a
=
d
k
if v = s(k)
d
k
if v = t(k)
0 else
for each v V, (1)
where δ
+
(v) and δ
(v) respectively denote the set of outgoing arcs and incoming arcs at node v.
In this work the values of the demand vector are uncertain and belong to the closed, convex, and
bounded set D R
|K|
+
. We call such a set an uncertainty set and any d D is called a realization
of the demand. We denote by (D, R
|A|×|K|
) the set of all functions from D to R
|A|×|K|
. Then, a
routing is a function f (D, R
|A|×|K|
) that satisﬁes (1) for all realizations of the demand, that is
X
aδ
+
(v)
f
k
a
(d)
X
aδ
(v)
f
k
a
(d) =
d
k
if v = s(k)
d
k
if v = t(k)
0 else
for all v V, d D, (2)
and that is non-negative
f
k
a
(d) 0 for all d D. (3)
2

A routing with no further restrictions is called dynamic routing. Hence, the set of all dynamic
routings is the set of all functions from D to R
|A|×|K|
that satisfy (2) and (3):
F
n
f (D, R
|A|×|K|
) | f satisﬁes (2) and (3)
o
. (4)
In this paper, we are interested in using special kinds of routings. This corresponds to using speciﬁc
subsets F
0
F. These subsets are described in the next section. In what follows, we describe the
robust network design problem making explicit the set of admissible routings.
A vector x R
|A|
+
is called a capacity allocation. A capacity allocation is said to support the set
D if there exists a dynamic routing f F serving D such that for every d D the corresponding
multi-commodity ﬂow f(d) does not exceed the capacities described by x. Similarly, we say that
(x, f ) supports D when both the routing f and the capacity allocation x are given. More generally,
we say that (x, F
0
) supports D when there exists a routing f F
0
such that (x, f ) supports D.
Given an uncertainty set D and a routing set F
0
F, robust network design now aims at providing
the cost minimal capacity allocation x such that (x, F
0
) supports D:
min
X
aA
κ
a
x
a
RND(F
0
) s.t. f F
0
(5)
X
kK
f
k
(d) x, d D (6)
x 0,
where κ
a
R is the cost for installing one unit of capacity on arc a A. Notice that in real
applications, these costs are usually non-negative. We shall denote the optimal cost of RND(F
0
)
by opt(F
0
). Problem RND(F
0
) contains an inﬁnite number of variables f (d) for all d D as well
as an inﬁnite number of capacity constraints (6). Moreover, the problem may not even be linear,
depending on the constraints deﬁning set F
0
.
Considering the set of all routings F, RND(F) is a two-stage robust program with recourse
following the more general framework described by Ben-Tal et al. [8]. The capacity design has to
be ﬁxed in the ﬁrst stage, and observing a demand realization d D, we are allowed to adjust the
routing f (d) arbitrarily in the second stage. In that case, (5) is replaced by (2) and (3) so that
RND(F) is a linear program, yet inﬁnite. Whenever D is a polytope, Poss and Raack [26], among
others, show how to provide a ﬁnite linear programming formulation for RND(F). The formulation
is based on enumerating the extreme points of D, so that its size tends to increase exponentially with
the number of commodities. In fact, the problem is very diﬃcult to solve given that only deciding
whether a given capacity allocation vector x supports D is coN P-complete for general polytopes D,
see Chekuri et al. [14] and Gupta et al. [18]. Moreover, the use of dynamic routings suﬀers from
another drawback. It may be diﬃcult in practice to change arbitrarily the routing according to the
demand realization.
For these reasons, various authors study restrictions on the routings that can be used, introducing
diﬀerent subsets of routings F
0
F. Their hope is that opt(F
0
) provides a good approximation of
opt(F) while yielding an easier optimization problem RND(F
0
). For instance, Frangioni et al. [16]
and Poss and Raack [26] show under very strong assumptions on D that the optimal capacity al-
locations provided by dynamic routings are equivalent to the ones provided by static routings and
aﬃne routings, respectively, which are polynomially solvable when D has a compact formulation. In
the next section, we present diﬀerent choices of F
0
discussed in the literature, including static and
aﬃne routings. Then, we summarize the main contributions of this paper in Section 2.3.
Note that if there exists only one path from s(k) to t(k) for a commodity k K, then all routings
coincide for that commodity. Unless stated otherwise, in the following we assume that for all k K
there exist at least two distinct paths p
1
, p
2
in G from s(k) to t(k), that is, two paths that diﬀer by
one arc at least.
2.2 Routings frameworks
In the next sections, we deﬁne formally the set of static routings and the routing sets from Ouorou
and Vial [24], Ben-Ameur [3], Scutella [27] and Ben-Ameur and Zotkiewicz [6].
3

2.2.1 Static routing
The simplest alternative to dynamic routing has been introduced by Ben-ameur [5] and has been
used extensively since then, see Altin et al. [1, 2], Koster et al. [19], Mudchanatongsuk et al. [21],
and Ord´nez and Zhao [22]. This framework considers a restriction on the second stage recourse
known as static routing (also called oblivious routing). Each component f
k
: D R
|A|
+
is forced to
be a linear function of d
k
:
f
k
a
(d) := y
k
a
d
k
a A, k K, d D. (7)
Notice that (7) implies that the ﬂow for k is not changing if we perturb the demand for h 6= k. By
combining (2) and (7) it follows that the multipliers y R
|A|×|K|
+
satisfy to
X
aδ
+
(v)
y
k
a
X
aδ
(v)
y
k
a
=
1 if v = s(k)
1 if v = t(k)
0 else
for each v V. (8)
The ﬂow y is called a routing template since it decides, for every commodity, which paths are used
to route the demand and what is the percental splitting among these paths. We deﬁne formally the
the set of all routing templates as
Y
n
y R
|A|×|K|
+
| y satisﬁes (8)
o
, (9)
and the set of all static routings as
F
stat
n
f (D, R
|A|×|K|
) | y Y : f
k
a
(d) = y
k
a
d
k
a A, k K, d D
o
.
An important result is that a compact linear formulation can be provided for RND(F
stat
) as long as
the description of D is compact (see Altin et al. [2] among others). Hence, the resulting optimization
problem is polynomially solvable.
In the following, we review alternative routing sets F
0
that are less restrictive than static routings
while not being as ﬂexible as dynamic routings. Said diﬀerently, F
stat
F
0
F.
2.2.2 Covers of the uncertainty set delimited by a hyperplane
Given a set D, a collection of subsets of D forms a cover of D if D is a subset of the union of sets in
the collection. Ben-Ameur [3] introduces the idea of covering the uncertainty set by two (or more)
subsets using hyperplanes and proposes to use a routing template for each subset. This yields the
following set of routings:
F
2|
n
f (D, R
|A|×|K|
) | y
1
, y
2
Y and α R
K
, β R :
f
k
a
(d) =
y
1k
a
d
k
d D {d, αd β}
y
2k
a
d
k
d D {d, αd β}
a A, k K, d D
.
The deﬁnition above implies that both routing templates y
1
and y
2
must be able to route demand
vectors that lie in the hyperplane {d, αd = β} without exceeding the capacity. He proves that
RND(F
2|
) is N P-hard in general and describes simpliﬁcation schemes, where α is given. He further
works on the framework in Ben-Ameur and Zotkiewicz [7].
2.2.3 Arbitrary covers of the uncertainty set
Scutella [27] introduces the idea of using conjointly two routing templates. Formally, she proposes
to use two routing templates y
1
and y
2
such that each d D can be served either by y
1
or by y
2
(or
both). This yields the following set of routings:
F
2
n
f (D, R
|A|×|K|
) | y
1
, y
2
Y and D
1
, D
2
D, D = D
1
D
2
:
f
k
a
(d) =
y
1k
a
d
k
d D
1
y
2k
a
d
k
d D
2
a A, k K, d D
.
4

She mentions that the complexity of RND(F
2
) is unknown. We show in this paper that this
optimization problem is N P-hard, because it is a generalization of RN D(F
2|
), proved to be N P-
hard by Ben-Ameur [3]. The framework described by F
2
has been independently proposed for general
robust programs by Bertsimas and Caramanis [11] (see also Bertsimas et al. [12]) where the authors
propose to cover the uncertainty sets with k subsets and devise independent sets of recourse variables
for each of these subsets.
2.2.4 Volume routings
More recently, Ben-Ameur and Zotkiewicz [6] introduce a framework that shares the demand between
two routing templates, according to thresholds h
k
for each k K. Formally, they use the following
set of routings:
F
V
n
f (D, R
|A|×|K|
) | y
1
, y
2
Y, h R
K
+
:
f
k
a
(d) = y
1k
a
min(d
k
, h
k
) + y
2k
a
max(d
k
h
k
, 0) a A, k K, d D
.
They prove that RND(F
V
) is an N P-hard optimization problem. Hence, they introduce simpler
frameworks described below. Deﬁning d
k
min
= min
d∈D
d
k
and d
k
max
= max
d∈D
d
k
, the set of routings
becomes one of the following
F
VS
n
f (D, R
|A|×|K|
) | y
1
, y
2
Y : f
k
a
(d) = y
1k
a
d
k
min
+ y
2k
a
(d
k
d
k
min
) a A, k K, d D
o
,
F
VG
n
f (D, R
|A|×|K|
) | y
1
, y
2
Y :
f
k
a
(d) = y
1k
a
d
k
min
d
k
max
d
k
d
k
max
d
k
min
+ y
2k
a
d
k
max
d
k
d
k
min
d
k
max
d
k
min
a A, k K, d D
,
which are both well-deﬁned whenever d
k
min
< d
k
max
for each k K. When d
k
min
= d
k
max
for some
k K, the k-th component of f F
VG
is deﬁned by f
k
(d) = y
1k
d
k
.
2.2.5 Aﬃne routings
Ben-Tal et al. [8] introduce Aﬃne Adjustable Robust Counterparts restricting the recourse to be an
aﬃne function of the uncertainties. Ourou and Vial [24] apply this framework to robust network
design by restricting f
k
to be an aﬃne function of all components of d giving
F
aff
{f (D, R
|A|×|K|
)|∃f
0
R
K
, y R
|A|×|K|
:
f
k
a
(d) = f
0k
a
+
X
hK
y
kh
a
d
h
a A, k K, d D, f satisﬁes (2) and (3)
)
.
This framework has been compared theoretically and numerically to static and dynamic routings by
Poss and Raack [26]. In particular, the authors show that a compact formulation can be described
for RN D(F
aff
) as long as D has a compact description, generalizing the result obtained for static
routing already. We point out that a major diﬀerence between F
aff
and the routing described in
Section 2.2.1-2.2.4 is that the formers are build up using routing templates, so that it is implicitly
assumed that ﬂow conservation constraints (2) and non-negativity constraints (3) are satisﬁed. In
opposition, routings in F
aff
are build up using ordinary vectors so that that satisfaction of (2) and
(3) must be stated explicitly.
2.3 Contributions of this paper
The objective of this paper is to compare opt(F
0
) among the routing sets recalled in previous sections.
This comparison is carried out in Section 3. Our main results are stated next.
(a) Let D be an uncertainty set. It holds that opt(F
2
) = opt(F
2|
), opt(F
aff
) opt(F
VG
)
opt(F
VS
), and opt(F
V
) opt(F
VS
) for any cost vector κ R
|A|
.
(b) Let D be an uncertainty polytope such that for each k K, there exists non-negative numbers
0 d
k
min
d
k
max
such that d
k
{d
k
min
, d
k
max
} for each extreme point of D. It holds that
opt(F
V
) = opt(F
VS
) for any cost vector κ R
|A|
.
5

##### Citations
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Journal ArticleDOI
Josette Ayoub
TL;DR: A general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty and shows that the relative performance of the algorithms depend on whether the budget is integer or fractional.
Abstract: We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to poly-tope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max-min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.

42 citations

### Cites background or methods or result from "A comparison of routing sets for ro..."

• ...Other approximations of (RND) have been considered in the literature, see Poss (2013) and the references therein....

[...]

• ...In the next section, we apply the two versions of Algorithm 1 to the instances studied by Poss and Raack (2013)....

[...]

• ...The above optimization problem has been studied previously by Poss and Raack (2013). Poss and Raack (2013) solves (RND) by enumerating the extreme points of Ξ using Lemma 1....

[...]

• ...…polytopes having limited number of extreme points and they address larger problems by applying affine decision rules to routing variables y: yk(ξ) = f0k + ∑ h∈K fhkξh, k ∈ K. (25) Other approximations of (RND) have been considered in the literature, see Poss (2013) and the references therein....

[...]

• ...We assess our algorithms on a difficult telecommunication network design problem that has previously been studied in the literature by Poss and Raack (2013), comparing our results with the affine decision rules....

[...]

Journal ArticleDOI
TL;DR: The expansion of a telecommunications network faces two sources of uncertainty, which are the demand for traffic that will transit through the expanded network and the outsourcing cost that the network operator will have to pay to handle the traffic that exceeds the capacity of her network.
Abstract: The expansion of a telecommunications network faces two sources of uncertainty, which are the demand for traffic that will transit through the expanded network and the outsourcing cost that the network operator will have to pay to handle the traffic that exceeds the capacity of her network. The latter is determined by the future cost of telecommunications services, whose negative correlation with the total demand is empirically measured in the literature through the price elasticity of demand.

13 citations

### Cites background from "A comparison of routing sets for ro..."

• ...Notice that Poss (2013, 2014b) has recently pointed out the conservatism of the second probabilistic bound from Theorem 5....

[...]

• ...Intermediary routing schemes have also been considered, see Poss (2014a) and the references therein, among which affine routing, introduced by Ouorou and Vial (2007), stands out for offering a good balance between computational complexity and flexibility....

[...]

Journal ArticleDOI
Sara Mattia
TL;DR: The polyhedral study provides evidence that some well-known valid inequalities are facets for all the considered routing/flows policies under the same assumptions, and introduces a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead, they are facets in some settings, but not in others.
Abstract: We consider the capacity formulation of the Robust Network Loading Problem. The aim of the paper is to study what happens from the theoretical and from the computational point of view when the routing policy (or scheme) changes. The theoretical results consider static, volume, affine and dynamic routing, along with splittable and unsplittable flows. Our polyhedral study provides evidence that some well-known valid inequalities (the robust cutset inequalities) are facets for all the considered routing/flows policies under the same assumptions. We also introduce a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead, they are facets in some settings, but not in others. A branch-and-cut algorithm is also proposed and tested. The computational experiments refer to the problem with splittable flows and the budgeted uncertainty set. We report results on several instances coming from real-life networks, also including historical traffic data, as well as on randomly generated instances. Our results show that the problem with static and volume routing can be solved quite efficiently in practice and that, in many cases, volume routing is cheaper than static routing, thus possibly representing the best compromise between cost and computing time. Moreover, unlikely from what one may expect, the problem with dynamic routing is easier to solve than the one with affine routing, which is hardly tractable, even using decomposition methods.

12 citations

### Cites background from "A comparison of routing sets for ro..."

• ...However, they lead to optimization problems that are even harder to solve than the problem with dynamic routing [43]....

[...]

Journal ArticleDOI
01 Jul 2018-Networks
TL;DR: The k‐adaptive routing scheme, based on the fact that the decision‐maker chooses k second‐stage solutions and then commits to one of them only after realization of the uncertainty, is experiment with for the robust network loading problem with demand uncertainty.
Abstract: We experiment with an alternative routing scheme for the robust network loading problem with demand uncertainty. Named k‐adaptive, it is based on the fact that the decision‐maker chooses k second‐stage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding k‐partition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control time performance. We compare our k‐adaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic networks from SNDlib and other sources.

7 citations

### Cites background or methods from "A comparison of routing sets for ro..."

• ...Also one can check that opt(Faff ) ≤ opt(FVA) ≤ opt(FVG) using the same arguments of [28]....

[...]

• ...In [28] the author compares optimal solutions for different routing strategies F, defined as opt(F)....

[...]

• ...We define each routing scheme from a robust optimization perspective as in [28]....

[...]

Journal ArticleDOI
TL;DR: The k-adaptive routing scheme, based on the fact that the decision-maker chooses k second-stage solutions and then commits to one of them only after realization of the uncertainty, is experimented for the Robust Network Loading problem.
Abstract: We experiment an alternative routing scheme for the Robust Network Loading problem. Named k-adaptive, it is based on the fact that the decision-maker chooses k second-stage solutions and then commits to one of them only after realization of the uncertainty. This routing scheme, with its corresponding k-partition of the uncertainty set, is dynamically defined under an iterative method to sequentially improve the solution. The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control its tractability. We compare our k-adaptive results with the ones obtained through other routing schemes and also verify the effectiveness of the methods utilized using several realistic instances from SNDlib.

3 citations

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### "A comparison of routing sets for ro..." refers background in this paper

• ...The simplest alternative to dynamic routing has been introduced by [13,14] and has been used extensively since then, see [5,1,18,20]....

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TL;DR: An approach is proposed that flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations, and an attractive aspect of this method is that the new robust formulation is also a linear optimization problem, so it naturally extend to discrete optimization problems in a tractable way.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

2,935 citations

### "A comparison of routing sets for ro..." refers background or methods in this paper

• ...Bertsimas and Sim [13] consider general linear programs where the coefficients of each linear inequality belong to intervals such that the number of coefficients taking conjointly their maximum value is bounded by a constant Γ....

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• ...Similarly, if dkmin = d k max, then any routing in FVS or FVG is uniquely determined by a unique routing template for k. Suppose that 0 dkmin d k max....

[...]

• ...The polytope introduced by Bertsimas and Sim [13], used for robust network design problems in [6, 23, 24, 20, 26], satisfies the assumption of (b) when the number of deviations allowed is integer....

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• ...Further works by Chen and Zhang [15] and Goh and Sim [17] suggest to extend the second stage to piece-wise linear functions of the uncertainties....

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• ...Similarly, flow f(d2) uses the routing template y 2k = fk(dk2)/d k 2 for k = k1, k2....

[...]

Journal ArticleDOI
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

2,874 citations

Journal ArticleDOI
TL;DR: It is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.
Abstract: We treat in this paper linear programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its robust counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.

1,672 citations

### "A comparison of routing sets for ro..." refers background or methods in this paper

• ...This result follows directly from the dualization technique introduced by [9] for robust linear optimization....

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• ...In this context, a solution is said to be feasible for the problem if it is feasible for all demand vectors that belong to the estimated uncertainty set D, see the seminal works of [26] and [9]....

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Journal ArticleDOI
TL;DR: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy via set containment.
Abstract: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$ \end{document} where the binary operation + refers to addition of sets. The problem is then to find x ∈ X that maximizes the linear function c · x. When the res...

1,575 citations

### "A comparison of routing sets for ro..." refers background in this paper

• ...In this context, a solution is said to be feasible for the problem if it is feasible for all demand vectors that belong to the estimated uncertainty set D, see Soyster [28] and Ben-Tal and Nemirovski [9, 10], among others....

[...]