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Journal ArticleDOI

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

23 Feb 2016-Computational Management Science (Springer Berlin Heidelberg)-Vol. 13, Iss: 2, pp 219-239
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.

Summary (2 min read)

1 Introduction

  • Robust Optimization is now a well-developed paradigm to tackle optimization problems under uncertainty.
  • In the robust counterpart of an optimization problem, the constraints involving uncertain parameters must be feasible for all values of the uncertain parameters in the convex sets.
  • In these problems, the demand of the customers is usually not known with precision until the links or the facilities are constructed.
  • The main advantage of affine decision rules is their tractability, since they lead to robust optimization problems with the structure of classical robust counterparts.
  • The idea behind RCG has also been combined with RG heuristically by Bertsimas et al (2013).

2 Problem overview

  • The authors follow a classical assumption from the robust optimization literature and suppose that T depends affinely on uncertain parameter ξ.
  • One readily sees that (P ) encompasses more general problems where (i) h depends affinely on ξ and (ii) second stage variables y have fixed costs given by vector k.
  • Similarly, the approaches presented in this paper can be applied (with minor modifications) to problems where set S is the intersection of Z|I1| × R|I2| and a polyhedron.
  • The ideas presented in this paper rely on considering only the extreme points of Ξ, which is formalized in the result below (whose proof can be found in Ben-Tal and Nemirovski (2002), among others).
  • (3) We provide next an example showing that Lemma 1 may not hold if the recourse matrix W depends on the uncertainty parameters ξ, which implies that their approach cannot handle problems with random recourse.the authors.the authors.

3 Solution approach

  • Then, assuming that the separation problem can be solved adequately, the authors describe in Section 3.2 two algorithms for addressing (P ′).
  • The next result is based on Farka’s lemma.
  • The drawback of RG is that a single cut is added to the master problem at each iteration, facing the risk that many iterations may be required before obtaining a feasible solution for (P ′).
  • For completeness, both algorithms are described formally in Algorithm 1.

4 Mixed-integer programming reformulations

  • The authors propose two mixed-integer linear reformulations for (SP ).
  • Notice that such a mapping always exists since the authors can represent Ξ as the set of all convex combinations of the vectors in vert(Ξ).
  • This example is useless, however, because the associated polytope Ω would be the unit simplex of dimension | vert(Ξ)|.
  • A different approach has been used by Mattia (2013) for a specific network design problem.
  • Ut Notice that in spite of its generality, the reformulation from Theorem 3 can be very hard to solve to optimality (Mattia, 2014).

5 Application to telecommunications network design

  • 1 Problem description Telecommunications networks have evolved quickly, especially in the last two decades.
  • Network operators are then under constant pressure to design high speed networks with larger capacities such as to satisfy the growing need for fast connections with no interruptions and latency.
  • In contrast Table 2 shows that RCG can solve only two instances with fractional values of Γ , while RG can solve roughly half of them.
  • The tables also show that both algorithms spend most of their time in the solution of (SPL) apart from RCG for the instances with high numbers of iterations.

6 Conclusion

  • The authors propose in this paper decomposition algorithms to solve adjustable robust linear programs under polytope uncertainty by generating the extreme points of the uncertainty polytope on the fly.
  • The authors discuss algorithmic frameworks for decomposing the robust problem and solution algorithms to address the subproblem.
  • The authors numerical results show that the relative efficiency of row generation and row-and-column generation algorithms depends on the type of polytope considered.
  • In any case, the bottleneck of these algorithms lies in finding the extreme point of the polytope that is mostly violated by the current solution, which amounts to solve a mixed-integer linear program (MIP).
  • The authors would like to thank Sara Mattia for fruitful discussions on the topic of this paper.

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Decomposition for adjustable robust linear optimization
subject to uncertainty polytope
Josette Ayoub, Michael Poss
To cite this version:
Josette Ayoub, Michael Poss. Decomposition for adjustable robust linear optimization subject to
uncertainty polytope. Computational Management Science, Springer Verlag, 2016, 13 (2), pp.219-
239. �10.1007/s10287-016-0249-2�. �hal-01301374�

Noname manuscript No.
(will be inserted by the editor)
Decomposition for adjustable robust linear
optimization subject to uncertainty polytope
Josette Ayoub · Michael Poss
Received: date / Accepted: date
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 gen-
eration 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.
Keywords Adjustable robust optimization · Uncertainty polytope · Benders
decomposition · Mixed-integer linear programming · Network design
1 Introduction
Robust Optimization is now a well-developed paradigm to tackle optimization
problems under uncertainty. The framework has experienced its revival in the
late nineties independently by Ben-Tal and Nemirovski (1998); El Ghaoui et al
(1998); Kouvelis and Yu (1997), and has witnessed an increasing attention in
J. Ayoub
CDS Consultant at Murex, Lebanon
E-mail: josetteayoub@hotmail.com
M. Poss
UMR CNRS 5506 LIRMM, Universit´e de Montpellier, rue Ada 161, 34095 Montpellier cedex
5, France
E-mail: michael.poss@lirmm.fr

2 Josette Ayoub, Michael Poss
the past twenty years. Its essence lies in the use of convex sets to model
uncertainty that can arise when solving optimization problems. In the robust
counterpart of an optimization problem, the constraints involving uncertain
parameters must be feasible for all values of the uncertain parameters in the
convex sets. In particular, the optimization variables are fixed independently of
the values taken by the uncertain parameters; it is not possible to adjust them
after the uncertainty is known. When the problem constraints are linear, this
approach leads to tractable optimization problems. For instance, the robust
counterparts of linear constraints subject to polyhedral uncertainty are still
linear constraints.
The framework can fail to model design problems that involve actions
that are delayed in time, such as network design problems or facility location
problems, among many others. In each of these optimization problem, we must
take part of the decisions today, e.g. implantation of new links or building
facilities. Then, when the new links or facilities are operational, we must choose
how to use them optimally to provide a service to the customers. In these
problems, the demand of the customers is usually not known with precision
until the links or the facilities are constructed.
Adjustable robust optimization has been introduced by Ben-Tal et al (2004)
to improve over static robust optimization by allowing a subset of variables
to account for the uncertainty. Namely, the framework partitions the opti-
mization variables into two sets: part of them must fix their values before the
uncertainty is revealed while the rest of them can adjust themselves accord-
ing to the values taken by the uncertain parameters. These variables become
functions defined on the uncertainty set. Ben-Tal et al (2004) prove that ad-
justable robust optimization is untractable in general, so that they focus on
approximations that restrict the adjustable variables to affine functions of the
uncertainty, yielding the so-called affine decision rules. The main advantage of
affine decision rules is their tractability, since they lead to robust optimization
problems with the structure of classical robust counterparts. The properties of
affine decision rules have been studied in subsequent papers: Bertsimas et al
(2011b) and Iancu et al (2013) study conditions in which affine decision rules
are optimal, while Bertsimas and Goyal (2012) study the suboptimality of
affine decision rules from a worst-case perspective. Authors have also studied
more complex decision rules that offer more flexibility than affine decision rules
while providing tractable optimization problems. Among others, Chen et al
(2008) introduce deflected linear and segregated linear decision rules, Chen and
Zhang (2009) propose to define affine decision rules on extended descriptions
of the uncertainty set, Goh and Sim (2010) introduce complex piece-wise lin-
ear decision rules defined through liftings and projections, and Bertsimas and
Georghiou (2015) propose piece-wise affine decision rules having fixed number
of affine pieces. Alternatively, Bertsimas and Caramanis (2010) dynamically
partition the uncertainty set and use constant decision rules in each set of
the partition. The performance guarantee of the latter scheme is studied the-
oretically by Bertsimas et al (2011a). This idea has been revived recently by
Bertsimas and Dunning (2014) and Postek and Den Hertog (2014) who extend

Decomposition for robust optimization 3
it to multi-stage linear mixed-integer linear programs and test it numerically
on different problems.
To assess numerically the quality of the aforementioned approximations,
one needs to compute the (exact) optimal solutions, at least on small instances.
Hence, in contrast with approximations approaches, some authors have tried
to solve exactly the adjustable problems. Indeed, when the robust constraints
are linear and the uncertainty set is a polytope, we know that the latter can
be replaced by the finite set of its extreme points. This reformulation as such
is not very useful because the number of extreme points is usually prohibitive.
However, recent works have proposed decomposition algorithms that gener-
ate the extreme points on the fly. The first work in that line of research was
carried out by Bienstock and
¨
Ozbay (2008) who propose cutting plane algo-
rithms (denoted RG in the following) for computing optimal base-stock levels
in a supply chain. Similar approaches have been used subsequently by Mat-
tia (2013) who study a network design problem with integer link capacities
under demand uncertainty, by Gabrel et al (2014) who study a facility loca-
tion problem under demand uncertainty, and by Bertsimas et al (2013) who
study a unit commitment problem under nodal injection uncertainty. In these
four papers, the uncertainty is limited to the right-hand side of the constraints
and problem specific algorithms are proposed. Zeng and Zhao (2013) improved
over the previous papers by proposing a row-and-column generation algorithm
(denoted RCG in the following) and comparing the latter numerically to RG.
Their results show that RCG can be up to three order of magnitudes faster
than RG. The idea behind RCG has also been combined with RG heuristically
by Bertsimas et al (2013). More general problems (based on algorithm RG)
have also been considered by Billionnet et al (2014).
This papers contributes to this line of research, proposing an alternative
approach to solve exactly two-stage robust linear programs with continuous
second stage variables. We implement cutting plane algorithms and row-and-
column generation algorithms very close to those proposed by Zeng and Zhao
(2013), with the difference that we consider the separation problem as a fea-
sibility problem instead of a min-max problem as done by Zeng and Zhao
(2013). Using the Farkas’ lemma, we can reformulate the feasibility problem
into a bilinear program. Whenever the uncertainty set can be obtained as an
affine projection of a 0 1 polytope, which is the case for the budgeted uncer-
tainty polytope from Bertsimas and Sim (2004), the bilinear program can be
linearized to obtain a mixed-integer linear program. 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.

4 Josette Ayoub, Michael Poss
2 Problem overview
We consider in this paper the following type of two-stage robust optimization
problems:
min c · x (1)
s.t. x S
(P ) T (ξ)x + W y(ξ) h, ξ Ξ (2)
where u · v denotes the scalar product between any pair of vectors u and v,
c R
|I|
, W R
|M|×|J|
, h R
|M|
, S R
|I|
denotes the first-stage feasibility
polyhedron, Ξ R
|K|
denotes the uncertainty polytope, T (ξ) R
|M|×|I|
denotes the realization of the uncertain first-stage coefficient matrix, and y(ξ)
denotes the second-stage decision vector. We follow a classical assumption
from the robust optimization literature and suppose that T depends affinely
on uncertain parameter ξ. Hence, there exists matrices T
0
and T
1k
with the
same dimensions as T such that
T (ξ) := T
0
+
X
k
T
1k
ξ
k
.
One readily sees that (P ) encompasses more general problems where (i) h
depends affinely on ξ and (ii) second stage variables y have fixed costs given
by vector k. Similarly, the approaches presented in this paper can be applied
(with minor modifications) to problems where set S is the intersection of
Z
|I
1
|
× R
|I
2
|
and a polyhedron.
When Ξ is not a singleton, problem (P ) is a linear program that contains
an infinite number of variables, since y is defined for each ξ Ξ, as well as
an infinite number of constraints (2). The ideas presented in this paper rely
on considering only the extreme points of Ξ, which is formalized in the result
below (whose proof can be found in Ben-Tal and Nemirovski (2002), among
others).
Lemma 1 Let vert(Ξ) be the set of extreme points of Ξ and x R
|I|
a given
vector. Vector x can be extended to an optimal solution (x, y) to (P ) if and
only if it can be extended to an optimal solution (x, y
0
) to
min c · x
s.t. x S
(P
0
) T (ξ)x + W y
0
(ξ) h ξ vert(Ξ). (3)
We provide next an example showing that Lemma 1 may not hold if the
recourse matrix W depends on the uncertainty parameters ξ, which implies
that our approach cannot handle problems with random recourse. Consider a
unique robust constraint defined by W (ξ) = 1 + 2ξ, T (ξ) = 2 + 3ξ, h = 0,

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"Decomposition for adjustable robust..." refers methods in this paper

  • ...Figure 2 shows performance profiles (Dolan and Moré, 2002) that compare the solution times of algorithms RG and RCG as well as affine routing (denoted aff)....

    [...]

  • ...Figure 2 shows performance profiles (Dolan and Moré, 2002) that compare the solution times of algorithms RG and RCG as well as affine routing (denoted aff)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
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.

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01 Jan 2004
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.

3,359 citations


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  • ...Whenever the uncertainty set can be obtained as an affine projection of a 0− 1 polytope, which is the case for the budgeted uncertainty polytope from Bertsimas and Sim (2004), the bilinear program can be linearized to obtain a mixed-integer linear program....

    [...]

  • ...Finally, the big-M coefficients can be set to 1, because 0 ≤ πm ≤ 1, which yields constraints (15) and (16). ut An interesting example of non-bijective affine mapping arises with the budgeted uncertainty set introduced in Bertsimas and Sim (2004): ΞΓ ≡ { ξ ∈ [0, 1]|K| s.t. ∑ k∈K ξk ≤ Γ } ....

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  • ...Our uncertainty set is based on the budgeted uncertainty polytope from Bertsimas and Sim (2004)....

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"Decomposition for adjustable robust..." refers background in this paper

  • ...The framework has experienced its revival in the late nineties independently by Ben-Tal and Nemirovski (1998); El Ghaoui et al (1998); Kouvelis and Yu (1997), and has witnessed an increasing attention in J. Ayoub CDS Consultant at Murex, Lebanon E-mail: josetteayoub@hotmail.com M. Poss UMR CNRS…...

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Abstract: Preface. 1. Approaches to Handle Uncertainty In Decision Making. 2. A Robust Discrete Optimization Framework. 3. Computational Complexity Results of Robust Discrete Optimization Problems. 4. Easily Solvable Cases of Robust Discrete Optimization Problems. 5. Algorithmic Developments for Difficult Robust Discrete Optimization Problems. 6. Robust 1-Median Location Problems: Dynamic Aspects and Uncertainty. 7. Robust Scheduling Problems. 8. Robust Uncapacitated Network Design and International Sourcing Problems. 9. Robust Discrete Optimization: Past Successes and Future Challenges.

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Frequently Asked Questions (10)
Q1. What have the authors contributed in "Decomposition for adjustable robust linear optimization subject to uncertainty polytope" ?

The authors present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Their 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 their 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, the authors can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. 

Their first mixed-integer linear formulation for (SP ) is based on reformulating Ξ as the affine mapping of a 0 − 1 polytope Ω ⊂ R|K′| ofreasonable dimension |K ′|. 

The main advantage of affine decision rules is their tractability, since they lead to robust optimization problems with the structure of classical robust counterparts. 

Chen et al (2008) introduce deflected linear and segregated linear decision rules, Chen and Zhang (2009) propose to define affine decision rules on extended descriptions of the uncertainty set, Goh and Sim (2010) introduce complex piece-wise linear decision rules defined through liftings and projections, and Bertsimas and Georghiou (2015) propose piece-wise affine decision rules having fixed number of affine pieces. 

Authors have also studied more complex decision rules that offer more flexibility than affine decision rules while providing tractable optimization problems. 

To assess numerically the quality of the aforementioned approximations, one needs to compute the (exact) optimal solutions, at least on small instances. 

CPLEX is called with its default parameters and the authors have set a time limit of T seconds of CPU time for every individual run and their computing times are presented in seconds. 

In any case, the bottleneck of these algorithms lies in finding the extreme point of the polytope that is mostly violated by the current solution, which amounts to solve a mixed-integer linear program (MIP). 

Vector x∗ belongs to K if and only if the optimal solution of the following optimization problem is non-positivemax (h− T̃ 0x∗) · π − ∑ k∈K (T̃ 1kx∗) · vk(SPL) s.t. ω ∈ Ω WTπ = 01 · π = 1 vkm ≥ πm − (1− ωk) k ∈ K ′,m ∈M (15) vkm ≤ ωk k ∈ K ′,m ∈M (16) π, vkm ≥ 0, ω ∈ {0, 1}|K ′|where T̃ 1k = ∑ h∈K T 1hAhk for each k ∈ K and T̃ 0 = T 0 + ∑ k∈K T 1k ∑ h∈K Akhb k.Proof 

An interesting example of non-bijective affine mapping arises with the budgeted uncertainty set introduced in Bertsimas and Sim (2004):ΞΓ ≡ { ξ ∈ [0, 1]|K| s.t.∑ k∈K ξk ≤