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Adaptive robust optimization with objective
uncertainty
Boris Detienne1, Henri Lefebvre2, Enrico Malaguti2
and Michele Monaci2*
1IMB UMR CNRS 5251, Inria Bordeaux Sud-Ouest, Universit´e de
Bordeaux, 200 Avenue de la Vieille Tour, Talence, 33405, France.
2*Dipartimento di Ingegneria dell’Energia Elettrica e
dell’Informazione “Guglielmo Marconi”, Universit`a di Bologna,
Viale del Risorgimento, 2, Bologna, 40136, BO, Italy.
*Corresponding author(s). E-mail(s): michele.monaci@unibo.it;
Contributing authors: boris.detienne@u-bordeaux.fr;
henri.lefebvre@unibo.it;enrico.malaguti@unibo.it;
Abstract
In this work, we study optimization problems where some cost param-
eters are not known at decision time and the decision flow is modeled
as a two-stage process within a robust optimization setting. We address
general problems in which all constraints (including those linking the
first and the second stages) are defined by convex functions and involve
mixed-integer variables, thus extending the existing literature to a much
wider class of problems. We show how these problems can be refor-
mulated using Fenchel duality, allowing to derive an enumerative exact
algorithm, for which we prove -convergence in a finite number of opera-
tions. An implementation of the resulting algorithm, embedding a column
generation scheme, is then computationally evaluated on two different
problems, using instances that are derived starting from the existing
literature. To the best of our knowledge, this is the first approach
providing results on the practical solution of this class of problems.
Keywords: two-stage robust optimization, reformulation, Fenchel duality,
column generation, branch-and-bound, computational experiments
1 Introduction
Robust Optimization (RO) has emerged as a solution approach to deal with
uncertainty in optimization problems. Contrary to stochastic optimization,
another popular approach that relies on probability distributions, robust opti-
mization considers an uncertainty set for the unknown parameters, against
which the taken decision should be immune. In that sense, constraints have
to be respected in every possible realization of the parameters and the objec-
tive function evaluated in the least advantageous case. The concept was first
introduced in Soyster (1973) and received considerable attention in the scien-
tific literature. Recent advances in RO can be found in Bertsimas et al (2010),
Hassene et al (2009), Ben-Tal et al (2009), Leyffer et al (2020) and Yanıko˘glu
et al (2019), among others.
More formally, a basic robust optimization problem can be cast as follows:
inf
z
z
zsup
ξ
ξ
ξ∈Ξ
f(ξ
ξ
ξ, z
z
z)
subject to g
g
g(ξ
ξ
ξ, z
z
z)≤0∀ξ
ξ
ξ∈Ξ
z
z
z∈Z
(1SR-P)
Here, the unknown data is represented by variables ξ
ξ
ξthat belong to the so-
called uncertainty set Ξ. As mentioned above, decision z
z
zhas to be feasible
in every possible occurrence of the uncertainty, hence robust solutions tend
to be overly conservative. To tackle this drawback, so-called adjustable robust
optimization Ben-Tal et al (2004), also known as two-stage robust optimization,
was introduced. As its name suggests, in a two-stage context, part of the
decisions are made in a here-and-now phase (i.e., before uncertainty reveals),
while recourse decisions can be taken in a wait-and-see phase (i.e., once the
actual values of the uncertain data are known) as an attempt to react to the
outcome of the uncertain process. Typically, the feasible space of (1SR-P) can,
indeed, be recast to embed a two-stage decision process by splitting variables z
z
z
in (x
x
x,y
y
y) and defining set Zas X ×Y accordingly. With the convention that the
minimum objective function value for an infeasible problem is +∞, a two-stage
robust problem can be formulated as follows:
inf
x
x
x∈X sup
ξ
ξ
ξ∈Ξ
inf
y
y
y∈Y(x
x
x,ξ
ξ
ξ)f(ξ
ξ
ξ, x
x
x,y
y
y) (2SR-P)
where Y(x
x
x,ξ
ξ
ξ) = {y
y
y:y
y
y∈ Y, g
g
g(ξ
ξ
ξ, x
x
x,y
y
y)≤0}, and g
g
g(ξ
ξ
ξ, x
x
x,y
y
y)≤0are the
so-called linking constraints. Set Xis now referred to as the first-stage feasi-
ble space. Given x
x
x∈ X and ξ
ξ
ξ∈Ξ, the corresponding second-stage feasible
space is Y(x
x
x,ξ
ξ
ξ), and the second-stage problem is inf{f(ξ
ξ
ξ, x
x
x,y
y
y) : y
y
y∈ Y(x
x
x,ξ
ξ
ξ)}.
It is known Ben-Tal et al (2004) that problems which can be cast as two-
stage robust problems often are at least NP-hard, even in the case where
first and second stage variables are continuous and all the involved functions
are linear. Several approaches have been developed to tackle this class of
problems. Assuming that the second stage is continuous and exhibits strong
duality, it can be replaced by its dual. This way, the inner maximization
problem can be reformulated using its epigraph, leading to a constraint-
generation algorithm in the spirit of Benders’ decomposition (see, e.g., Terry
et al (2009), Bertsimas et al (2013), Jiang et al (2014) and Gabrel et al (2011)).
A column-and-constraint-generation scheme has been proposed in Zeng and
Zhao (2013), which consists in adding one set of recourse decision variables
and the corresponding second-stage constraints associated with a realization
of the uncertainty. These realizations are dynamically generated by solving a
bilevel problem. Later, the same approach was used in Ayoub and Poss (2016),
where the constraint-generation problem was modelled as a mixed integer pro-
gram exploiting a description of the uncertainty set in terms of its extreme
points. Note that this method can handle mixed-integer second-stage decisions,
which is not the case for classical Benders-type approaches. Unfortunately,
this method seems to be of practical relevance only when a small number of
variables has to be added for reaching optimality.
The inherent difficulty of this class of problems motivated the development
of approximate solution methods. In the affine decision rule approach proposed
in Ben-Tal et al (2004), the recourse decisions are expressed as affine functions
of the uncertainty. Another relevant approach, introduced in Bertsimas and
Caramanis (2010), is the finite adaptability (also known as K-adaptability) in
which the number of recourse decisions is restricted to some finite number. An
MILP formulation for the case of binary second-stage decisions and objective
uncertainty was proposed in Hanasusanto et al (2015) and a branch-and-bound
algorithm was later proposed in Subramanyam et al (2019) to address cases
with uncertain linear constraints.
An important special case of (2SR-P) arises when uncertainty affects the
objective function only, i.e., Y(x
x
x,ξ
ξ
ξ) = Y(x
x
x),∀ξ
ξ
ξ∈Ξ. For this specific case,
K¨ammerling and Kurtz (2020) proposed an oracle-based algorithm relying on
a hull relaxation combining the first- and second-stage feasible spaces embed-
ded within a branch-and-bound framework. However, this approach applies to
purely binary variables and linear constraints only. On the other hand, Arslan
and Detienne (2021) proposed an exact MILP reformulation of the problem
in case of linear linking constraints that involve binary variables only. Besides
solving the problem by means of a branch-and-price algorithm, a further con-
tribution of Arslan and Detienne (2021) is proving the NP-completeness of the
problem in this setting.
In the setting where uncertainty affects the objective function only, our
analysis shows that further effort is needed to tackle with more general cases,
in particular when linking constraints are defined by nonlinear functions or
involve both integer and continuous variables. Similarly, to the best of our
knowledge, the case in which the objective function is nonlinear has not
been considered yet. This paper contributes in filling this gap, as we con-
sider two-stage robust problems with objective uncertainty, convex constraints
and mixed-integer first and second stage. By extending in a non-trivial way
some recent results from the two-stage stochastic optimization literature (see
Sherali and Fraticelli (2002), Sherali and Zhu (2006) and Li and Grossmann
(2019)), we obtain a relaxation of the problem, and analyze its tightness for
different special cases. This relaxation can be embedded within a branch-and-
bound scheme thus producing an exact solution approach, for which we prove
finite ε-convergence. Besides the theoretical analysis, we also show that, from
a computational viewpoint, the proposed algorithm is able to solve instances
of practical relevance arising from two different applications. We also point out
that the class of problems which can be addressed by our solution approach
is quite large since we only require mild assumptions on the nature of the
involved optimization problem.
The article is organized as follows. In Section 2we formally introduce the
class of problems we are considering throughout this work, whereas Section
2.2 describes in greater details the algorithmic solution approach proposed
in Arslan and Detienne (2021) for a special case of our problem. In Section
3we present a relaxation of the problem, and an effective algorithm for its
solution. We then derive sufficient conditions for the relaxation to coincide
with the original problem in a mixed-integer context. In the purely binary case,
the equivalence between problem (2SRO-P) and our lower-bounding problem
is established without any condition. We then present a branch-and-bound
algorithm able to close the optimality gap with finite ε-convergence assuming
that the lower-bounding problem can be finitely solved with εtolerance. In
section 3.4, we propose a column-generation algorithm to solve the lower-
bounding problem with such property. Finally, section 4applies the proposed
algorithm to two problems: a capital budgeting problem and a capacitated
facility location problem.
Notations Throughout this paper, matrices and vectors are written in bold
case, e.g., x
x
x∈Rnor A
A
A∈Rn×m, while components are written in normal font,
e.g., xior aij . Columns of A
A
Aare written in bold case with exponent indexing,
e.g., a
a
ai. Let f:Rn→Rbe a given function with dom(f) = {x
x
x∈Rn:f(x
x
x)<
+∞}; its convex conjugate is denoted by f∗:Rn→Rand is given by
f∗(π
π
π) = sup
x
x
x∈dom(f){π
π
πTx
x
x−f(x
x
x)}
Similarly, we denote by f∗the concave conjugate of f. Let X⊆Rn×{0,1}n−p
be a given set, we denote Xits continuous relaxation, i.e., X=X∩Rn×
{0,1}n−p, conv (X) its convex hull, i.e., the smallest convex set Csatisfying
X⊆C.
The indicator function of Xis noted δ(·|X) and equals zero if its argument
belongs to Xand +∞otherwise. Its convex conjugate is therefore given by
δ∗(π
π
π|X) = sup{π
π
πTx
x
x:x
x
x∈X}. If Xis a convex polytope, we note vert(X)
the set of its extreme points. Finally, for a logical proposition E, function 1
1
1(E)
equals one if Eis true and zero otherwise.
2 Problem description
2.1 General setting
As anticipated, our goal is to solve problem (2SR-P) with objective uncertainty,
convex constraints and mixed-integer first and second stages.
For the sake of clarity, let us first introduce several sets. Set I={1, . . . , n1}
denotes the set of indices for the first-stage variables, and is partitioned into
two sets IBand IC: variables whose index belongs to IBare required to take
binary values, while those whose index belongs to ICare continuous variables,
i.e., wlog, X ⊂ R|IC|×{0,1}|IB|. Similarly, we introduce set J={1, . . . , n2}as
the indices for the second-stage variables and partition this set into JBand JC,
i.e., wlog, Y ⊂ R|JC|× {0,1}|JB|. Sets IBand JB, which correspond to binary
first and second-stage binary variables, may be defined in the same way in
case of general integer variables; all the results presented in the paper directly
extend to the integer case as well. Finally, we introduce set U={1, . . . , n3}
as the index set for the uncertain variables, i.e., Ξ ⊂Rn3.
We now explicit some assumptions on the problem.
Assumption 1 (Objective uncertainty) For all ξ
ξ
ξ∈Ξand x
x
x∈ X ,Y(ξ
ξ
ξ, x
x
x) = Y(x
x
x).
Assumption 2 (Convexity)
1. Xis compact and convex;
2. The uncertainty set Ξis a finite-dimensional, bounded convex set;
3. For all x
x
x∈ X,Y(x
x
x)is a finite-dimensional, bounded convex set;
4. The objective function fis a concave function of the uncertainty and a
convex function of the first- and second-stage decisions, i.e., fx
x
x,y
y
y:ξ
ξ
ξ7→
f(ξ
ξ
ξ, x
x
x,y
y
y)is a concave function for all fixed x
x
x∈ X and y
y
y∈ Y(x
x
x)and
fξ
ξ
ξ: (x
x
x,y
y
y)7→ f(ξ
ξ
ξ, x
x
x,y
y
y)is a convex function for all fixed ξ
ξ
ξ∈Ξ.
Assumption 3 (Complete recourse) For every (relaxed) first-stage decision, there
exists at least one feasible second-stage decision, i.e., for every x
x
x∈X,Y(x
x
x)is a
non-empty set.
Assumption 4 (Boundedness)
1. The objective function fis bounded over the first- and second-stage feasible
space, i.e., for all fixed ξ
ξ
ξ∈Ξ,{(x
x
x,y
y
y) : x
x
x∈ X, y
y
y∈ Y(x
x
x)} ⊆ dom (fξ
ξ
ξ)
2. For all (x
x
x,y
y
y) : x
x
x∈ X and y
y
y∈ Y(x
x
x),relint(Ξ) ∩dom (fx
x
x,y
y
y)6=∅
Assumption 5 (Separability) Let Q={1,...,q}.
1. The objective function fcan be expressed as a sum of qfunctions, i.e.,
there exist qfunctions (ψi:R|U|+|I|+|J|→R)i∈Qsuch that f(ξ
ξ
ξ, x
x
x,y
y
y) =
Pi∈Qψi(ξ
ξ
ξ, x
x
x,y
y
y)for all x
x
x∈ X, y
y
y∈ Y(x
x
x)and all ξ
ξ
ξ∈Ξ.
2. For all i∈Q,ψiis separable in ξ
ξ
ξand (x
x
x,y
y
y)meaning that there exists
functions (wi:R|U|→R)i∈Qand (ϕi:R|I|+|J|→R)i∈Qsuch that
ψi(ξ
ξ
ξ, x
x
x,y
y
y) = wi(ξ
ξ
ξ)ϕi(x
x
x,y
y
y). In addition, we assume that wi(·)is a concave
function and ϕi(·)is a convex function.
A few remarks regarding these assumptions are necessary. First, note that
Assumptions 1and 2are here to define what we refer to as convex mixed-
integer robust problems with objective uncertainty. It is important to highlight
that the word ”convex” is here to suggest that all involved functions are convex
with respect to the first- and second-stage variables. Yet, in general, even
under these assumptions, problem (2SR-P) may fail to have a straightforward
convex MINLP formulation, as function h:x
x
x7→ maxξ
ξ
ξ∈Ξminy
y
y∈Y(x
x
x)f(ξ
ξ
ξ, x
x
x,y
y
y)
is not necessarily a convex function over the continuous relaxation of X. We
give here a small example.
Example 1 (nonconvex MINLP) Consider the following first- and second-stage
feasible regions:
X= [0,1] and Y(x) = y
y
y∈ {0,1}2
y1+y2≤1
y1≤1−x
By inspection, we have that (y1, y2) = (0,0) and (y1, y2) = (0,1) are always feasible
second-stage solutions, while (y1, y2) = (1,0) is feasible only when x= 0. Fixing
the uncertainty set Ξ = [0,1], we take interest in the following convex mixed-integer
two-stage robust problem:
min
x∈[0,1] h(x) with h:x7→ max
ξ∈[0,1] min
(y1,y2)∈Y(x)ξ(−2y1+y2+ 1)
Though every involved functions are convex (in fact, affine), the following holds:
h(x) =
max
ξ∈[0,1] min {ξ; 2ξ;−ξ}= 0 if x= 0
max
ξ∈[0,1] min {ξ; 2ξ}= 1 if x > 0= 1
1
1(x > 0)
Clearly, hfails to be convex over [0,1] which ends our example.
Assumption 3is a standard assumption in the two-stage optimization liter-
ature, and is known to be easy to enforce as soon as the considered problem is
bounded, which is implied by Assumption 4.1. Assumption 4.2 is not restrictive
in practice, and will be used in the proof of lemma (2).
Finally, Assumption 5is structural to our work, and implies the following
remark.
Remark 1 Without loss of generality, for all i∈Q, we can assume that ϕi(·) is a
convex function (at most affine).
Proof Let i∈Qsuch that ϕi(·) is concave, then, to fulfill Assumption 2.1, wi(ξ
ξ
ξ)
must be negative forall ξ
ξ
ξ∈Ξ. Thus, one may equivalently replace wi(·) by −wi(·)
and ϕi(·) by −ϕi(·).
Remark 2 For all i∈Qsuch that ϕi(·) (resp. wi(·)) is not single-signed, then wi(·)
(resp. ϕi(·)) is affine.
Remark 3 For all i∈Qsuch that ϕi(·) (resp. wi(·)) is not affine, then wi(·) (resp.
ϕi(·)) is a non-negative function.
Note that Assumption 5could be relaxed to address situations in which,
for i∈Qsuch that ϕi(·) (resp. wi(·)) is affine, there is no restriction on the
concavity (resp. convexity) of the associated wi(·) (resp. ϕi(·)).
Example 2 (Fulfilling Assumption 5) We give here some examples of functions which
satisfy Assumption 5. For simplicity, we denote z
z
z= (x
x
x, y
y
y).
•Uncertain linear functions of the form (ξ
ξ
ξ, z
z
z)7→ ξ
ξ
ξA
A
Az
z
zwhere A
A
Ais a given real
matrix;
•Diagonal uncertain convex quadratic form (ξ
ξ
ξ, z
z
z)7→ z
z
zTdiag(ξ
ξ
ξ)z
z
zwhere ξ
ξ
ξ≥0;
•Uncertain positively weighted sum of convex functions of the form (ξ
ξ
ξ, z
z
z)7→
Pi∈Qξiϕi(z
z
z) with Ξ ⊂R|U|
+, e.g., (ξ
ξ
ξ, x
x
x,y
y
y)7→ Pi∈Qξix2
i/yiwith y
y
y≥0.
Example 3 (Violating Assumption 5) We give here some examples of functions which
do not satisfy Assumption 5.
•Non-concave functions of the uncertainty, e.g., (ξ
ξ
ξ, z
z
z)7→ ||z
z
z−ξ
ξ
ξ|| for any given
norm;
•General uncertain quadratic form (Σ
Σ
Σ,z
z
z)7→ z
z
zTΣ
Σ
Σz
z
zeven with Σ
Σ
Σ0 (unless
Ξ∩R|U|
−=∅)
In the following lemma, we finally state the class of problems we consider.
Lemma 1 Under Assumptions (1)-(5), there exists [l
l
l, u
u
u]⊂R|I|+|J|such that
(2SR-P)is equivalent to the following problem:
inf
x
x
x∈X ∩[l
l
l,u
u
u]sup
ξ
ξ
ξ∈Ξ
inf
(t
t
t,y
y
y)∈Y0(x
x
x)X
i∈Q
wi(ξ
ξ
ξ)ti(2SRO-P)
with Y0(x
x
x)such that Y(x
x
x) = projy
y
y(Y0(x
x
x)) and Y0(x
x
x)is a convex and finite-
dimensional set.
Proof The existence of the hyper-rectangle [l
l
l, u
u
u] is trivial as Xis assumed to be
bounded (Assumption 2.1). Moreover, the following equality holds:
inf
y
y
y
X
i∈Q
wi(ξ
ξ
ξ)ϕi(x
x
x, y
y
y) : y
y
y∈ Y(x
x
x)
= inf
y
y
y,t
t
t
X
i∈Q
wi(ξ
ξ
ξ)ti:y
y
y∈ Y(x
x
x), ti=ϕi(x
x
x, y
y
y),∀i∈Q
However, the optimization problem on the right side of the equality may fail to be
convex if there exists i∈Qsuch that ϕiis not affine. Let QA⊆Qbe the set of
indices for which ϕiis affine. By Assumption 5, for all i∈Q\QA, we have wi(·)≥0
and thus constraint ”ti=ϕi(x
x
x, y
y
y)” may be equivalently replaced by ”ti≥ϕi(x
x
x, y
y
y)”,
which is convex. We therefore can choose
Y0(x
x
x) =
(t
t
t, y
y
y) :
y
y
y∈ Y(x
x
x)
ti=ϕi(x
x
x, y
y
y)∀i∈QA
ti≥ϕi(x
x
x, y
y
y)∀i∈Q\QA
For every x
x
x∈X, the continuous relaxation of Y0(x
x
x) is convex and non-empty
(Assumption 3); by construction, it is also finite dimensional.
In what remains, we will assume to know a hyper-rectangle [l
l
l, u
u
u] as
described in lemma 1.
2.2 Special case: linear and binary setting
We complete the introduction by discussing the special case of (2SRO-P) under
the following additional assumptions:
1. X,Ξ and x
x
x7→ Y(x
x
x) are defined by linear constraints;
2. there exists a matrix A
A
A∈R|U|×|Q|such that ∀i∈Q, wi(ξ
ξ
ξ) = ξ
ξ
ξTa
a
ai; and
3. linking constraints are defined by functions g(x
x
x,y
y
y) that do not depend on
first-stage variables in Ic.
In a recent paper Arslan and Detienne (2021), the authors observed that,
for this variant of the problem, the inner minimization miny
y
y∈Y(x
x
x)ξ
ξ
ξTA
A
Ay
y
ycan be
equivalently replaced by miny
y
y∈conv(Y(x
x
x)) ξ
ξ
ξTA
A
Ay
y
y, i.e., the second-stage feasible
space can be substituted by its convex hull. This allows to transform the min-
max-min problem into a min-max problem using the well known minimax
theorem for convex sets. Assuming that Ξ is expressed as {ξ
ξ
ξ∈R|U|
+:F
F
Fξ
ξ
ξ≤
d
d
d}, the inner maximization problem is dualized so as to obtain the following
equivalent problem:
min
x
x
x,y
y
y,λ
λ
λd
d
dTλ
λ
λ(1)
subject to x
x
x∈ X (2)
y
y
y∈conv (Y(x
x
x)) (3)
F
F
FTλ
λ
λ≥A
A
Ay
y
y(4)
λ
λ
λ≥0,(5)
where λ
λ
λare the dual variables associated to the inner maximization problem.
Note that, besides the integrality requirements on the variables, the only non-
convex constraint is (3). By exploiting a reformulation already used in Sherali
and Fraticelli (2002), Sherali and Zhu (2006) and Li and Grossmann (2019)
for two-stage stochastic optimization problems with mixed-integer first and
second stage, Arslan and Detienne (2021) showed that this constraint may be
equivalently enforced as
(x
x
x,y
y
y)∈conv (S)∩ {(x
x
x0,y
y
y0) : x
x
x=x
x
x0},(6)
where S={(x
x
x,y
y
y) : x
x
x∈ {0,1},y
y
y∈ Y(x
x
x)}. The obtained reformulation is then
solved by means of a branch-and-price algorithm where the convex hull of
set Sis expressed in terms of convex combinations of its extreme points, and
branching is performed on the first-stage variables only.
3 A hull-relaxation-based branch-and-bound
algorithm
In this section we present our main contribution and its theoretical founda-
tions. In the same spirit as in Arslan and Detienne (2021), we first turn problem
(2SRO-P) from a min-max-min problem to a min-max problem in our mixed-
integer and convex context. Then, since linear duality does not apply in our
setting, we resort to Fenchel duality to obtain a reformulation of the problem.
Similarly to the linear and binary case, we then replace the counterpart of
constraint (3) by (6), though this only provides a relaxation of the problem in
the general setting. This relaxation is thus embedded into a branch-and-bound
scheme to obtain an optimal solution of (2SRO-P).
3.1 Problem reformulation
The following lemma extends to the mixed-integer and convex context the
result given in Arslan and Detienne (2021).
Lemma 2 (Single-stage reformulation) Problem (2SRO-P)is equivalent to the
following problem:
inf
(x
x
x,t
t
t,y
y
y)∈Fsup
ξ
ξ
ξ∈ΞX
i∈Q
wi(ξ
ξ
ξ)ti(7)
with F={(x
x
x, t
t
t, y
y
y) : x
x
x∈ X ∩ [l
l
l, u
u
u],(t
t
t, y
y
y)∈conv Y0(x
x
x)}.
Proof This lemma relies on the same arguments as those employed in Arslan and
Detienne (2021): first, the feasible space of the inner minimization problem is replaced
by its convex hull by linearity of the objective function and convexity of the feasible
region. By assumption 2.2 and Lemma 2, both Ξ and conv Y0(x
x
x)(for all x
x
x∈ X )
are convex and finite dimensional. Thus, the result in Perchet and Vigeral (2015)
can be used to turn the inner sup −inf into an inf −sup problem. This achieves the
proof.
The inner maximization problem may be turned into a minimization prob-
lem by use of Fenchel duality, as done in Ben-Tal et al (2009). In the following
proposition, we therefore derive a general reformulation of problem (2SRO-P).
Proposition 1 (Deterministic reformulation) Problem (2SRO-P)is equivalent to
the following problem:
inf
x
x
x,y
y
y,t
t
t,(v
v
vi)i∈Q,ξ
ξ
ξδ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi(8)
subject to x
x
x∈ X ∩ [l
l
l, u
u
u] (9)
(t
t
t, y
y
y)∈conv Y0(x
x
x)(10)
X
i∈Q
v
v
vi=ξ
ξ
ξ(11)
v
v
vi∈R|U|∀i∈Q(12)
Proof By a direct application of Fenchel duality and some conjugate calculus results,
the following holds
sup
ξ
ξ
ξ∈ΞX
i∈Q
tiwi(ξ
ξ
ξ) = sup
ξ
ξ
ξ∈R|U|
X
i∈Q
tiwi(ξ
ξ
ξ)−δ(ξ
ξ
ξ|Ξ)
= inf
ξ
ξ
ξ∈R|U|
δ∗(ξ
ξ
ξ|Ξ) −
X
i∈Q
tiwi(ξ
ξ
ξ)
∗
= inf
ξ
ξ
ξ∈R|U|
δ∗(ξ
ξ
ξ|Ξ) −sup
v
v
vi∈R|U|,i∈Q
X
i∈Q
(tiwi)∗v
v
vi:X
i∈Q
v
v
vi=ξ
ξ
ξ
= inf
δ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi:X
i∈Q
v
v
vi=ξ
ξ
ξ, v
v
vi∈R|U|, i ∈Q, ξ
ξ
ξ∈R|U|
See also appendix Afor more details on conjugate calculus.
Remark 4 Assume wlog that |Q|=|U|. If, for all i∈Q,wi(ξ
ξ
ξ) = wi(ξi), then problem
(2SRO-P) is equivalent to
inf
(x
x
x,t
t
t,y
y
y)∈F
δ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗(ξ
ξ
ξ)
(13)
Remark 5 Let i∈Qsuch that wi(·) is affine, i.e., wi(ξ
ξ
ξ)=(r
r
ri)Tξ
ξ
ξ+ri0. Problem
(2SRO-P) is equivalent to
inf
(x
x
x,t
t
t,y
y
y)∈Fnδ∗(R
R
Rt
t
t|Ξ) + r
r
rT
0t
t
to(14)
Proof Indeed, we have
(tiwi)∗(v
v
v) = inf
ξ
ξ
ξ∈R|U|{v
v
vTξ
ξ
ξ−ti((r
r
ri)Tξ
ξ
ξ+ri0)}=(−tiri0if v
v
v=tir
r
ri
−∞ otherwise.
These results show that although the reformulation for the general case
adds |Q| × |U|continuous variables, for some relevant cases these additional
variables can be omitted. In particular this is true in case all the wi(·) functions
are either separable or affine.
3.2 Relaxation
Note that the deterministic reformulation presented above still is not, in gen-
eral, a convex MINLP and that no tractable, compact form is known in the
general case. To overcome this drawback, we replace constraint (10) by the
following requirement:
(x
x
x,t
t
t,y
y
y)∈conv (S)∩ {(x
x
x0,t
t
t0,y
y
y0) : x
x
x=x
x
x0}
with S=
(x
x
x,t
t
t,y
y
y) :
lj≤xj≤uj∀j∈I
xj∈ {0,1} ∀j∈IB
(t
t
t,y
y
y)∈ Y0(x
x
x)
(15)
The problem obtained from this substitution is thus
min
x
x
x,y
y
y,(v
v
vi)i∈Q,ξ
ξ
ξδ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi
subject to x
x
x∈ X ∩ [l
l
l, u
u
u]
(x
x
x,t
t
t,y
y
y)∈conv (S)
X
i∈Q
v
v
vi=ξ
ξ
ξ
v
v
vi∈R|U|∀i∈Q
ξ
ξ
ξ∈R|U|
(P)
It is clear that, for any fixed ¯
x
x
x∈ X, we have {¯
x
x
x}×Y0(¯
x
x
x) = S∩ {(x
x
x,t
t
t,y
y
y) :
x
x
x=¯
x
x
x}, and that the same holds even for ¯
x
x
x∈ X. However, as shown, e.g., in
Sherali and Zhu (2006), the convexified counterpart does not hold, in the sense
that the inclusion ”{¯
x
x
x}×conv (Y(¯
x
x
x)) ⊆conv (S)∩{(x
x
x,t
t
t,y
y
y) : x
x
x=¯
x
x
x}” may be
strict. Example 4below illustrates this case.
Example 4 (Hull relaxation) We consider the first- and second-stage feasible sets
introduced in Example 1. In Figure (1a), we represent the convex hull of S. For a
fixed first-stage decision ¯x(here, ¯x= 0.4), Figure (1b) reports the feasible points for
constraint (15), whereas Figure (1c) describes the exact shape of conv (Y(¯x)). The
figure shows an example in which inclusion is strict. In addition, note that, whenever
¯xattains its bounds (i.e., ¯x∈ {0,1}), {¯x} × conv (Y(¯x)) = conv (S)∩ {(x, y
y
y) : x= ¯x}
holds.
The following Lemma follows from the considerations above.
y1
y2
x
(a) conv (S)
y1
y2
x
(b) conv (S)∩ {x= ¯x}
y1
y2
x
(c) {¯x} × conv (Y(¯x))
Fig. 1: Graphical representation of different sets from example 1
Lemma 3 (Lower-bounding property) Denoting by v(•)the optimal objective value
of problem •, we have:
v(P)≤v(2SRO-P)
In other words, (P) is a relaxation of (2SRO-P). In the next proposition,
we introduce a condition under which a feasible solution for problems (P) is
feasible for problem (2SRO-P) as well.
Proposition 2 If ¯
x
x
x∈vert ([l
l
l, u
u
u]), then
{¯
x
x
x} × conv Y0(¯
x
x
x)= conv (S)∩ {(x
x
x, t
t
t, y
y
y) : x
x
x=¯
x
x
x}
Proof Let ¯
x
x
x∈vert ([l
l
l, u
u
u]) and let (ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y)∈conv (S)∩ {(x
x
x, t
t
t, y
y
y) : x
x
x=¯
x
x
x}. Then,
(ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y) can be expressed as a (finite) convex combination of points of conv(S)
(Carath´eodory’s theorem), i.e.,
(ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y) = X
e∈E
(¯
xe,¯
te,¯
ye)αe
where Eis a given index list of such elements of conv (S). Assume that there exists
j∈Iand i∈Esuch that ¯xi
j6= ¯xj. If ¯xi
j>¯xj, condition ¯
xi∈conv (S) implies that
¯xj=lj. Hence, αi= 0 since ¯xk
j≥lj∀k∈E. The same argument shows that ¯xi
j<¯xj
implies αi= 0. Thus, for each e∈Esuch that αe>0, we must have ¯
xe=¯
x
x
x. This
implies that (¯
te,¯
ye)∈ Y0(¯
x
x
x) and thus Pe∈E(¯
te,¯
ye)αe∈conv Y0(¯
x
x
x).
Corollary 1 (Tightness condition) Let X∗be the set of optimal first-stage decisions
of problem (P). Then:
X∗∩vert ([l
l
l, u
u
u]) 6=∅ ⇒ v(P) = v(2SRO-P)
Proof Let (x
x
x∗, t
t
t∗, y
y
y∗) be an optimal solution of (P) with x
x
x∗∈vert ([l
l
l, u
u
u]). From
Proposition 2, it is also feasible for problem (2SRO-P). Thus, Lemma 3implies
optimality for problem (2SRO-P).
This result directly implies Corollary 2which states that, in the special
case where the first-stage variables are all binary, problem (P) is always an
exact reformulation of (2SRO-P).
Corollary 2 (Tightness condition/binary case) If the first-stage decisions are all
binary, i.e., IC=∅, then
v(P) = v(2SRO-P)
Proof In this case, [l
l
l, u
u
u] = [0,1], hence any optimal first-stage solution x
x
x∗satisfies
x
x
x∗∈ {0,1}IB= vert ([l
l
l, u
u
u]) which, by Corollary 1, proves the result.
3.3 Enumerative algorithm
We now present an exact method for solving problem (2SRO-P). Motivated by
Corollary 1, the main idea of the algorithm is to determine an optimal value
of the first-stage variables, and then derive the corresponding optimal values
for the second-stage variables. To this aim, we developed a branch-and-bound
algorithm in which we relax both the integrality of the xand requirement (10).
To ensure feasibility, we perform a spatial branching on the xvariables, until
each of them attains either its lower or upper bound. The algorithm stores the
best feasible solution found (the incumbent solution) which is returned when
the algorithm stops.
3.3.1 Node solution
Let pdenote a generic node of the branch-and-bound tree, associated with
bounds l
l
lpand u
u
upon first-stage variables.
A lower bound on the optimal solution value of node pcan be computed
solving the following problem:
min
x
x
x,t
t
t,y
y
y,(v
v
vi)i∈Q,ξ
ξ
ξδ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi
subject to x
x
x∈ X ∩ [l
l
lp,u
u
up]
(x
x
x,t
t
t,y
y
y)∈conv (Sp)
X
i∈Q
v
v
vi=ξ
ξ
ξ
v
v
vi∈R|U|∀i∈Q
ξ
ξ
ξ∈R|U|
(LBp)
where Sp={(x
x
x,t
t
t,y
y
y) : l
l
lp≤x
x
x≤u
u
up, xj∈ {0,1},∀j∈IB,(t
t
t,y
y
y)∈ Y0(x
x
x)}. This
problem is exactly the continuous relaxation of problem (P) where the bounds
l
l
land u
u
uhave been replaced by l
l
lpand u
u
up. Note that at the root node we have
l
l
l0=l
l
land u
u
u0=u
u
u.
Let (x
x
xp∗,t
t
tp∗,y
y
yp∗,(v
v
vip∗)i∈Q,ξ
ξ
ξp∗) be an optimal solution of problem LBp.
If v(LBp) is greater than or equal to the cost of the incumbent, the node is
fathomed by bounding. Otherwise, we distinguish three cases:
•if x
x
xp∗∈vert ([l
l
lp,u
u
up]), by Proposition 2, this solution is optimal for the
current node. Hence, the node is fathomed by optimality and the incumbent
is updated;
•if x
x
xp∗∈ X \ vert ([l
l
lp,u
u
up]), we compute a feasible solution for (2SRO-P) by
solving the following model in which the first-stage variables are fixed to x
x
xp∗:
min
t
t
t,y
y
y,(v
v
vi)i∈Q,ξ
ξ
ξδ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi
subject to (t
t
t,y
y
y)∈conv (Y0(x
x
xp∗))
X
i∈Q
v
v
vi=ξ
ξ
ξ
v
v
vi∈R|U|∀i∈Q
ξ
ξ
ξ∈R|U|
(UBp)
Note that, in this case, x
x
xp∗corresponds to a feasible first-stage solution;
hence, by Assumption 3, problem UBpis always feasible, and possibly the
incumbent is updated. If v(LBp) = v(U Bp) then node pis solved; otherwise,
we perform a branching;
•if x
x
xp∗∈ X \ X , we branch.
In the last case, before branching, one can try to round x
x
xp∗; if the result-
ing point is in X, a feasible solution for (2SRO-P) can be computed. In our
experiments, every fractional value for xp∗
jwith j∈IBwas rounded to the
closest integer while variables xp∗
jwith j∈ICwere not rounded.
3.3.2 Branching
We now describe how to select the branching variable at node p. For each first-
stage variable, say with index j∈I, we compute the minimum distance of xp∗
j
from one of its bounds at the node, i.e., we evaluate:
θp
j= min{xp∗
j−lp
j;up
j−xp∗
j}.
For branching, we give priority to binary variables that do not attain their
bound. Otherwise, we resort to spatial branching on continuous variables.
In both cases, we select the variable with maximum θp
jvalue, i.e., we select
variable xjsuch that,
j∈(argmax{θp
j:j∈IB}if ∃j∈IB, θj>0
argmax{θp
j:j∈IC}otherwise.
If j∈IB, then a standard binary branching is executed. Otherwise, we
resort to spatial branching, and generate two descendant nodes by imposing
y1
y2
x
(a) Left child (x≤β)
y1
y2
x
(b) Right child (x≥β)
Fig. 2: Branching on continuous variable xfrom example 1
xj≤xp∗
jfor the left node and xj≥xp∗
jfor the right one. We associate to
each node the lower bound value of the current node v(LBp) and insert them
in a list of open nodes. At each iteration, we extract from the list one node
with minimum lower bound value, halting the algorithm stops when the list is
empty.
Example 5 Figure 2illustrates the left and right child obtained by spatial branching
on x≤βand x≥βfrom example 1(here, β= 0.4). Clearly, the right child allows the
same recourse decisions as in Y(x) for all x≥β. The left child, however, still allows
second-stage decisions that could end up being infeasible in the original problem.
In particular, (x
x
x, y
y
y)=(ε, 1−ε, 0) with ε∈(0, β ] is feasible for (LBp) but not for
(2SRO-P).
3.3.3 Single-stage heuristic
We now present a heuristic procedure that can be used at the root node to
warm start the branch-and-bound algorithm. This heuristic is based on the
definition of a single-stage version of (2SR-P), in which both first- and second-
stage variables are simultaneously optimized. The resulting problem can be
formulated using the following MINLP:
min δ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi(16)
subject to x
x
x∈ X ∩ [l
l
l, u
u
u] (17)
(t
t
t,y
y
y)∈ Y0(x
x
x) (18)
X
i∈Q
v
v
vi=ξ
ξ
ξ(19)
v
v
vi∈R|U|∀i∈Q(20)
Note that solving this problem is NP-hard. Let x
x
x∗denote its optimal first-
stage solution with associated (t
t
t∗,y
y
y∗) second-stage solution. An improving
second stage-solution can be possibly obtained by fixing x
x
x=x
x
x∗in problem
(2SRO-P), in the spirit of the upper bounding procedure used when solving
UBp.
3.3.4 Convergence
Given a feasible solution of (LBp), say (x
x
x,t
t
t,y
y
y, V
V
V ,ξ
ξ
ξ), we introduce the following
function
F(x
x
x,t
t
t,y
y
y, V
V
V ,ξ
ξ
ξ) := δ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi=F(t
t
t,V
V
V ,ξ
ξ
ξ),
that returns the solution value in the lower bounding problem.
Proposition 3 (Convergence result) If F∈C0, our branch-and-bound algorithm
either finitely terminates or enters an infinite sequence of nodes for which the optimal
solutions of the associated lower bounding problems converge to an optimal solution
of 2SRO-P.
Proof Let us consider the case in which the algorithm enters an infinite sequence
Pof nodes, indexed by p. We denote by (l
l
lp, u
u
up) the associated bounds for the x
x
x
variables, and by (x
x
xp∗, t
t
tp∗, y
y
yp∗, V
V
Vp∗, ξ
ξ
ξp∗) the optimal optimal solutions of the corre-
sponding lower bounding problems. Since branching always reduces the domain of
the x
x
xvariables, then (l
l
lp, u
u
up) will converge to some values, say (l
l
l∗, u
u
u∗), and x
x
xp∗will
converge to a solution x
x
x∗.
We first show that the sequence of optimal solutions of the lower bounding prob-
lems converges to an optimal solution of the lower bounding problem defined by
bounds (l
l
l∗, u
u
u∗).
By boundedness of problems (LBp), there exists P0⊆Pand (t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗), such
that {(x
x
xp∗, t
t
tp∗, y
y
yp∗, V
V
Vp∗, ξ
ξ
ξp∗)}p∈P0→(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗)
Since (x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗) is the limit of a sequence of feasible points and the fea-
sible region is closed, (x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗) is feasible for the lower bounding problem
LB∗defined by bounds (l
l
l∗, u
u
u∗), and thus
F(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗)≥v(LB∗) := F(ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y, ˆ
V
V
V , ˆ
ξ
ξ
ξ),
where (ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y, ˆ
V
V
V , ˆ
ξ
ξ
ξ) is an optimal solution for problem LB∗. We now show that
(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗) is as well an optimal solution for this problem, i.e., equality holds.
Assume by contradiction that F(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗)> F (ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y, ˆ
V
V
V , ˆ
ξ
ξ
ξ). For all p∈P0,
since P0⊆P, we have [l
l
l∗, u
u
u∗]⊆[l
l
lp, u
u
up] and thus (ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y, ˆ
V
V
V , ˆ
ξ
ξ
ξ) is feasible for (LBp).
By continuity of Fwe now have
{F(x
x
xp∗, t
t
tp∗, y
y
yp∗, V
V
Vp∗, ξ
ξ
ξp∗)}p∈P0→F(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗)> F (ˆ
x
x
x,ˆ
t
t
t, ˆ
y
y
y, ˆ
V
V
V , ˆ
ξ
ξ
ξ),
which contradicts the optimality of (x
x
xp∗, t
t
tp∗, y
y
yp∗, V
V
Vp∗, ξ
ξ
ξp∗) for some p.
We now show that the solution to which the sequence converges is a feasible
solution for 2SRO-P. Given the infinite sequence of nodes, there exists at least one
variable j∈ICwhich is infinitely selected for branching. Thus, we must have θj→
0. Given our branching rule, this implies that all the other continuous variables
attain either their lower or upper bounds. Thus x
x
x∗∈vert ([l
l
l∗, u
u
u∗]), which implies
(t
t
t∗, y
y
y∗)∈conv Y0(x
x
x∗)and F(x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗) = v(LB∗)≥v(2SRO-P). Since our
node selection strategy always picks a node with minimum lower bound, for each node
pof the branching sequence we have v(LBp)≤v(2SRO-P)≤v(LB∗). As v(LBp)
converges to v(LB∗), we also have v(LB∗) = v(2SRO-P).
The previous result applies when considering infinite precision. By intro-
ducing a finite tolerance in the algorithm, we can show that an infinite
branching cannot occur at any point. More specifically, given an optimal solu-
tion x
x
xp∗of the lower bounding problem at a node p, we introduce the following
function
G(x
x
xp∗) := v(UBp),
that returns the solution value in the upper bounding problem.
Proposition 4 (Convergence in finite precision) If the branch-and-bound algorithm
enters an infinite sequence of nodes, then it must be converging to a point x
x
x∗in which
function Ghas a discontinuity.
Proof The proof of the previous proposition shows that there exists a subsequence
P0⊆Pthat converges to a solution (x
x
x∗, t
t
t∗, y
y
y∗, V
V
V∗, ξ
ξ
ξ∗). As (t
t
t∗, y
y
y∗)∈conv Y0(x
x
x∗),
then v(LB∗) = G(x
x
x∗).
Assume now that Gis continuous at x
x
x∗. Then, we have {G(x
x
xp∗)}p∈P0→G(x
x
x∗) =
v(LB∗), which allows us to fathom the node by optimality after a finite number of
nodes for any positive tolerance.
We conclude this section by observing that, at each node of the branch-and-
bound algorithm, the lower bounding problem can be solved with ε-tolerance
in a finite number of operations. Indeed, as shown in Ceria and Soares (1999)
and Grossmann and Ruiz (2012), one can reformulate a convex disjunctive
program as a compact convex MINLP by introducing an exponential number
of auxiliary variables that model the disjunctions. The resulting model can
thus be solved in finite number of states by using any algorithm designed for
convex optimization.
3.4 A convexification scheme based on column-generation
In this section, we propose a nonlinear column-generation algorithm to be used,
at each node p, to solve problem (LBp) to ε-optimality in a finite number of
iterations. According to this scheme, we approximate conv (Sp) by the convex
hull of a finite set of points belonging to Sp.
Restricted Master Problem: To determine this set, we use an iterative
approach. At each iteration k, let K={1, . . . , k}and denote by Hpk =
{(¯
xpj ,¯
tpj ,¯
ypj ) : j∈K}the associated set of points. We clearly have
conv Hpk⊆conv (Sp), thus the optimal solution of the problem obtained
by substituting conv (Sp) with conv Hpk in (LBp) gives an upper bound
of (LBp). The resulting problem, denoted as ( c
LBpk), is called the Restricted
Master, and is formulated as follows:
min
x
x
x,t
t
t,y
y
y,V
V
V ,ξ
ξ
ξ,α
α
αδ∗(ξ
ξ
ξ|Ξ) −X
i∈Q
(tiwi)∗v
v
vi(22)
subject to x
x
x∈ X ∩ [l
l
lp,u
u
up] (23)
x
x
x=X
j∈K
αj¯
xpj (24)
t
t
t=X
j∈K
αj¯
tpj (25)
y
y
y=X
j∈K
αj¯
ypj (26)
X
j∈K
αj= 1 (27)
X
i∈Q
v
v
vi=ξ
ξ
ξ(28)
v
v
vi∈R|U|∀i∈Q(29)
ξ
ξ
ξ∈R|U|(30)
αj≥0∀j∈K(31)
(c
LBpk)
Following the classical column-generation framework, the current approx-
imation can be improved by means of a so-called Pricing Problem, defined as
follows.
Pricing Problem: Let λ
λ
λpk∗, µ
µ
µpk∗, π
π
πpk∗and ηpk∗be the values of the dual
variables associated with constraints (24), (25), (26), and (27) in an optimal
solution of problem ( c
LBpk).
Pricing asks to solve the following problem
(¯
xp,k+1,¯
tp,k+1,¯
yp,k+1)∈argmin
(x
x
x,t
t
t,y
y
y)∈Sp−λ
λ
λpk∗Tx
x
x−µ
µ
µpkTt
t
t−π
π
πpk∗Ty
y
y−ηpk∗T(PPpk )
and generates a new point (¯
xp,k+1,¯
tp,k+1,¯
yp,k+1) belonging to Sp. If
v(PPpk)≥ −ε, we have an ε-optimal solution to (LBp), and hence the algo-
rithm terminates. Otherwise, we set Hk+1 =Hk∪ {(¯
xp,k+1,¯
tp,k+1,¯
yp,k+1)},
k=k+ 1 and iterate. Note that, at each iteration k, a lower bound on the
optimal solution value of (LBp) is given by v(c
LBpk)−v(PPpk ). This lower
bound, combined with an upper bound, can allow us to early terminate the
solution of problem (LBp).
The convergence of nonlinear column generation has been established in
Garc´ıa et al (2003) and implies finite ε-convergence of our method.
4 Computational experiments
In this section, we report computational results of our solution algorithm when
applied to two different optimization problems, a facility location problem and
a capital budgeting problem, respectively. Both problems are relevant from
an application viewpoint and are defined as non-trivial variants of problems
already addressed in the literature.
All the experiments were run on an AMD 3960 running at 3.8 GHz, with
a time limit equal to 3,600 CPU seconds per run.
4.1 Facility location problem with adjustable capacity
and set-up costs
We consider a company which has to decide, among a set V1of possible loca-
tions, the sites where to open a facility in order to serve a set V2of customers,
each with an associated with a known demand dj. The size of each opened
facility has to be determined as well, and an upper limit qion the capacity
that can be installed on each site i∈V1is given. The objective of the problem
is to minimize a cost function that includes both facility-opening costs and the
transportation costs to serve the customers. More into details, transportation
costs consist of a fixed component hij to be paid if customer jis assigned to
facility i, and a variable component ξij to be paid for each unit of good that
traverses this connection. We assume that the variable transportation costs
are not known precisely, i.e., ξij are uncertain parameters.
To model the problem, we introduce, for each location i∈V1, a binary
variable xitaking the value 1 if a facility is opened in location iand 0 other-
wise. In addition, we introduce a continuous variable zi∈[0,1] indicating the
fraction of the maximum capacity that is installed. Using these variables, the
first-stage feasible space is given by X={(x
x
x,z
z
z)∈ {0,1}|V1|×[0,1]|V1|:z
z
z≤x
x
x}.
To model the second-stage feasible region, we introduce, for every connection
between i∈V1and j∈V2, a binary variable yij which takes the value 1
if and only if the connection is used, and a continuous variable wij ∈[0,1]
representing the fraction of demand djserved by facility i. We then have:
Y(z
z
z) =
(y
y
y, w
w
w) :
y
y
y∈ {0,1}|V1|×|V2|
w
w
w∈[0,1]|V1|×|V2|
Pi∈V1wij = 1 ∀j∈V2
Pj∈V2djwij ≤ziqi∀i∈V1
wij ≤yij ∀i∈V1,∀j∈V2
(31)
Here, the first set of constraint enforces that every customer is served, while the
second set of constraint imposes that the installed capacities are not exceeded.
Finally, the last set of constraints link the wand the yvariables.
For each site i∈V, the associated opening cost depends on the activation
of the facility and on the installed capacity, and is described by the following
convex quadratic function
Fi(xi, zi) = fixi+γiz2
i+βizi
Our two-stage robust problem reads:
min
(x
x
x,z
z
z)∈X max
ξ
ξ
ξ∈Ξmin
(y
y
y,w
w
w)∈Y(z
z
z)X
i∈V1
fixi+γiz2
i+βizi+X
j∈V2
(hij yij +djξij wij )
(32)
We refer to problem (32) as the two-stage robust facility location problem with
adjustable capacity and set-up transportation costs. This problem can be cast
as (2SRO-P), where the linking constraints between the first- and second-
stage variables involve purely continuous variables and exhibits convexity in
the first-stage feasible space. Finally, the uncertainty has a linear impact on
the second-stage objective function.
We assume that variable transportation costs are described according to a
classical ellipsoidal uncertainty set
Ξκ=(ξij )i∈V1,j∈V2: (ξ
ξ
ξ−¯
ξ
ξ
ξ)Σ
Σ
Σ−1(ξ
ξ
ξ−¯
ξ
ξ
ξ)≤κ2,(33)
where κis a fixed sensitivity parameter, ¯
ξ
ξ
ξthe expected value of ξij and Σ
Σ
Σ =
(σij )i,j the covariance matrix. For simplicity, we will assume that Σ
Σ
Σ is diagonal
(implying that variable costs are independent from each other), although this
assumption is not strictly required by our method.
By applying Proposition 1, the following deterministic reformulation
results:
min
x
x
x,z
z
z,y
y
y,w
w
wX
i∈V1
fixi+γiz2
i+βizi+X
j∈V2
(hij yij +dj¯
ξij wij )
+κsX
i∈V1X
j∈V2
d2
jσ2
ij w2
ij (34)
subject to (x
x
x,z
z
z)∈ X (35)
(y
y
y, w
w
w)∈conv (Y(x
x
x)) (36)
The reader is referred to Appendix Bfor the derivation of the robust
counterpart of ellipsoidal uncertainty sets.
4.1.1 Instance generation
To test the proposed method, a large benchmark of random instances has been
generated in the spirit of Cornuejols et al (1991). First, we randomly generate
the location of each site i∈V1and customer j∈V2using a uniform distribu-
tion in the unit square. For each site iand customer j, the fixed component
of the transportation cost is obtained by multiplying the euclidean distance
between iand jtimes a random number with uniform distribution in [20,
50]. The expected value of the variable transportation cost is instead obtained
by multiplying the euclidean distance times 10. Each covariance parameter is
uniformly randomly generated in the interval [σ−, σ+], where σ−and σ+are
specified below. For every site i∈V1, the capacity upper limit is generated
according to a uniform distribution in [10,160], and the fixed setup cost is a
random number in [0,90] multiplied by a varying adjustment factor µ. Coef-
ficients γiand βiare computed so that Fi(1, qi), which represents the cost
for activating the full capacity on the site, equals √qi×α, where αis ran-
domly generated in [100,110], as in Cornuejols et al (1991). Finally, customers’
demands are obtained by defining a random vector d
d
din [0, 1], and scaling its
entries so that Pj∈V2qj/Pi∈V1di=ν, where νis another parameter.
In all our instances the number of sites and customers are
(4,8),(6,12),(8,16),(9,18),(10,20),(11,22) and (12,24), and the ellipsoidal
uncertainty parameter κbelongs to {1.0,1.5,2.0}. Covariance parameters σ−
and σ+take values (0.1,1),(0.5,2) and (1,4). Similar to Cornuejols et al
(1991), the adjustment factor µis set to 2.0 for instances with |V1||V2| ≤ 500
and to 1.0 otherwise, and νtakes values in {1.5,2.0,3.0,5.0,10.0}. Finally, for
each combination of the parameters we generate 4 instances, overall producing
1260 instances.
4.1.2 Numerical results
We now report the computational results of our branch-and-bound algorithm
without and with the use, at the root node, of the single-stage heuristic (SSH)
of Section 3.3.3; from now on, these versions are denoted as BB and BB H,
respectively. Table 1gives the outcome of our experiments for both variants
of the algorithm. Each line refers to the 60 instances characterized by the
same values of |V1|,|V2|and κ. Entries of the table give, for each variant, the
following information:
•nodes is the average number of branch-and-bound nodes explored (with
respect to instances solved to optimality only);
•time is the average computing time (with respect to instances solved to
optimality only);
•# opt is the number of optimal solutions.
In addition, for variant BB H, we report in column “% gapr” the average per-
centage gap at the root node. Letting Lrand Urbe the best lower and upper
bound at the root node, the gap is computed as % gapr= 100 ∗Ur−Lr
Lr.
This information is omitted for algorithm BB, which never provides a feasible
solution at the root node.
The table shows that the complexity of these ACFL instances increases with
the size of the underlying network. In addition, for each size of the network,
increasing the value of κ(i.e., allowing for more uncertainty in the realization
of the profits) makes the instances consistently harder. The basic algorithm BB
BB BB H
V 1 V 2 κnodes time # opt % gaprnodes time # opt
4 8 1 4.7 1.3 60 19.6 4.4 1.2 60
1.5 4.9 1.5 60 19.0 4.5 1.5 60
2 5.0 2.0 60 18.8 4.8 2.0 60
6 12 1 8.2 15.1 60 19.9 7.6 12.1 60
1.5 8.9 17.9 60 19.5 8.2 16.0 60
2 9.1 22.5 60 19.4 8.4 17.7 60
8 16 1 13.8 128.1 60 17.8 12.4 77.3 60
1.5 13.6 195.2 60 17.4 12.8 113.4 60
2 13.6 199.2 59 17.1 12.9 191.9 60
9 18 1 17.4 390.3 58 16.5 15.3 296.9 59
1.5 18.2 441.4 57 17.0 15.4 281.4 56
2 18.2 392.0 53 17.2 15.9 232.2 54
10 20 1 21.2 824.2 53 18.2 21.3 544.7 57
1.5 24.0 845.8 50 18.0 21.5 669.2 56
2 24.6 843.6 48 18.4 21.6 526.3 52
11 22 1 31.1 1369.0 49 20.9 28.6 873.2 55
1.5 32.0 1229.6 41 20.7 26.9 773.2 51
2 33.4 1265.3 37 21.2 29.1 1113.0 49
12 24 1 31.9 1325.2 33 19.1 25.2 1113.1 42
1.5 33.2 1195.4 30 20.4 27.4 1138.3 38
2 33.5 1208.9 27 20.3 28.0 994.9 34
Table 1: Performance of different variants of the algorithm in solving ACFL
problem
solves 85% of the instances, with an average computing time below 10 minutes.
Adding SSH heuristic produces a considerable improvement of the results:
although the average root node gap is around 20%, the algorithm solves 68
additional instances to optimality (more than 90% in total) and has an average
computing time which is reduced by more than 20%.
4.2 Robust Capital Budgeting problem
Our second test-case is a variant of the Robust Capital Budgeting (RCB)
problem introduced in Hanasusanto et al (2015) and considered also in Arslan
and Detienne (2021).
Consider a company which can allocate a given budget Bto a set of projects
i∈ N ={1, . . . , N }; the budget can be increased with loans. Each project
i∈ N has a fixed cost ciand an uncertain profit ˜pi(ξ
ξ
ξ) which depends on M
unknown factors ξ
ξ
ξthat belong to an uncertainty set Ξ. The company must
decide which projects should be activated to maximize the expected profit. To
this aim, it may activate some projects after observing the risk factors, though
late investments are less effective and are discounted by a factor f∈[0,1)
of their value. In addition, the company has the possibility to request loans
both in the first and in the second stage. The maximum amount of a loan is
denoted by C1and C2, respectively, and the interest rate is denoted by λand
is increased in the second stage by a factor µ.
The problem can be modeled as the following max-min-max problem.
max
(x
x
x,x0)∈X min
ξ
ξ
ξ∈Ξmax
(y
y
y,y0)∈Y(x
x
x,x0)"X
i∈N
˜pi(ξ
ξ
ξ)(xi+fyi)−(1 + λ)C1x0−(1 + λµ)C2y0#
(37)
with X={(x
x
x, x0)∈ {0,1}N×[0,1] : Pi∈N cixi≤B+C1x0}and Y(x
x
x, x0) =
{(y
y
y, y0)∈ {0,1}N×[0,1] : Pi∈N ci(xi+yi)≤B+C1x0+C2y0, yi+xi≤
1∀i∈ N}. Here, x
x
xare binary variables that indicate whether a project has
been activated in the first stage or not, while y
y
yindicate their activation in
the second stage. Variable x0(resp. y0) is a continuous variable indicating the
fraction of the loan capacity which is activated in the first (resp. second) stage.
The actual profit associated with each project is given by ˜pi(ξ
ξ
ξ) = ¯pi(1 +
∆i(ξ
ξ
ξ)) where ¯piis the nominal profit, ξ
ξ
ξdenotes the uncertainty belonging
to set Ξ = [−1,1]M, and ∆i: [−1,1]M→[−0.5,0.5] is a quadratic convex
function defined as ∆i(ξ
ξ
ξ) = ξ
ξ
ξTQ
Q
Qiξ
ξ
ξ/2 + g
g
giTξ
ξ
ξ.
4.2.1 Convex reformulation
In this section, we show how one can apply Proposition 1to reformulate (37).
By convexifying the inner maximization problem and swapping the inner max
and min operators, one obtains
max −(1 + λ)C1x0−(1 + λµ)C2y0+X
i∈N
¯pi(xi+fyi)
+ min
ξ
ξ
ξ∈ΞX
i∈N
∆i(ξ
ξ
ξ)(xi+fyi)¯pi(38)
subject to (x
x
x, x0)∈ X (39)
(y
y
y, y0)∈conv (Y(x
x
x, x0)) .(40)
We now reformulate the minimization subproblem as follows
min
ξ
ξ
ξ∈ΞX
i∈N
∆i(ξ
ξ
ξ)(xi+fyi)¯pi= min
ξ
ξ
ξ∈RM"X
i∈N
∆i(ξ
ξ
ξ)(xi+fyi)¯pi+δ(ξ
ξ
ξ|Ξ)#(41)
= max
ξ
ξ
ξ∈RM(−δ)∗(ξ
ξ
ξ|Ξ) − X
i∈N
∆i(ξ
ξ
ξ)(xi+fyi)¯pi!∗
(42)
= max (−δ)∗(ξ
ξ
ξ|Ξ) −X
i∈N ti∆i(s
s
si)∗
subject to ti= (xi+f yi)¯pi∀i∈ N
X
i∈N
s
s
si=ξ
ξ
ξ
ξ
ξ
ξ∈RM
s
s
si∈RM∀i∈ N
t
t
t∈R|N |
+
(43)
Note that (−δ)∗(ξ
ξ
ξ|Ξ) = −δ∗(−ξ
ξ
ξ|Ξ). In addition, by symmetry of Ξ, we
have δ(−ξ
ξ
ξ|Ξ) = δ(ξ
ξ
ξ|Ξ), hence δ∗(−ξ
ξ
ξ|Ξ) = δ∗(ξ
ξ
ξ|Ξ) = maxζ
ζ
ζ∈Ξξ
ξ
ξTζ
ζ
ζ=||ξ
ξ
ξ||1,
where the last equivalence is based on strong linear duality.
By combining these two results, we have that (−δ)∗(ξ
ξ
ξ|Ξ) = −||ξ
ξ
ξ||1.
Moreover
∆∗
i(ξ
ξ
ξ) = 1
2(ξ
ξ
ξ−g
g
gi)TQ
Q
Qi−1(ξ
ξ
ξ−g
g
gi) (44)
hence expanding each conjugate that appears in the summation in (43), we get
(ti∆i)∗(s
s
si) =
s
s
siQ
Q
Qi−1s
s
si
2ti−(g
g
giTQ
Q
Qi−1)Ts
s
si+ (g
g
giTQ
Q
Qi−1g
g
gi/2)tiif ti>0
0 if ti= 0 and s
s
si=0
+∞otherwise
(45)
As these terms are minimized in the objective function, any optimal solu-
tion with ti= 0 must have s
s
si= 0 as well. Note that this can be enforced
by introducing, for each project i∈N, a non-negative variable riand the
additional constraint
2tiri≥s
s
siTQ
Q
Qi−1s
s
si.
This constraint is convex since it can be modeled by means of the rotated
quadratic cone.
Our final model reads
max (46)
−(1 + λ)C1x0−(1 + λ)µC2y0+X
i∈N "¯pi(xi+fyi)− |ξi| − ri
+(g
g
giTQ
Q
Qi−1)Ts
s
si− g
g
giTQ
Q
Qi−1g
g
gi
2!ti#(47)
subject to (48)
(x
x
x, x0)∈ X (49)
(y
y
y, y0)∈conv (Y(x
x
x, x0)) (50)
ti= (xi+fyi)¯pi∀i∈ N (51)
2tiri≥s
s
siTQ
Q
Qi−1s
s
si∀i∈ N (52)
X
i∈N
s
s
si=ξ
ξ
ξ(53)
ξ
ξ
ξ∈R|N | (54)
s
s
si∈R|N | ∀i∈ N (55)
t
t
t∈R|N |
+(56)
r
r
r∈R|N |
+(57)
4.2.2 Instance generation
The instances in our testbed are generated similar to those in Arslan and
Detienne (2021): for each project i∈ N,ciis randomly generated following a
uniform distribution between 1 and 100, and the nominal profit ¯piis defined
as ci/5. We set the investment budget B=HPi∈N ciwhere His a given
parameter. We assume that postponed investments are discounted by a factor
f= 0.8.
The loans limits are set to C1=C2= 0.2Bwhile the interest rate
parameters are chosen as λ= 0.025 and µ= 1.2.
Deviations of the profits are generated as follows. For each project i, we
first randomly generate Mcoefficients g
g
gitaken taken from the M-dimensional
unit simplex, and then each such coefficient is multiplied by -0.5 or 0.5 with
equal probability. Then, we compute the value of the linear function g
g
giT ξ
ξ
ξin all
vertices of [−1,1]M, and compute a convex quadratic function that (i) has the
same value as the linear function on each vertex, and (ii) attains its minimum
(over [−1,1]M) in one of these points. (More details about this last step are
given in the Appendix).
In our instances, parameter Htakes values in {0.2,0.4,0.6,0.8}, the number
of projects Nis in {30,40,50,60}while the number of risk factors Mis either
4 or 8. For each of these settings, we define 10 instances, making a total of 320
instances.
4.2.3 Numerical results
Table 2gives the same information as Table 1. Since for this problem there are
cases in which algorithm BB provides a feasible solution at the root node, we
also give column “% gapr” (computed as % gapr= 100 ∗Ur−Lr
Ur) and column
“# UBr” reporting the number of instances for which this happens. This latter
column is omitted for algorithm BB H, as the SSH heuristic is always able to
determine a feasible solution at the root node. Each entry in the table refers
to the 40 instances characterized by the same value of Nand M, with the
exception of those in column % gapr, which account only the instances for
which a valid upper bound Uris available.
The results show that algorithm BB is able to solve almost all the instances
in at most 3 minutes. Algorithm BB H has slightly better performances, as
it allows to solve to optimality one more instance and reduces the average
computing time and number of nodes with respect to BB. Overall, BB H is able
to solve all instances but 2 to proven optimality, within less than one minute,
on average. Finally, observe that BB H may present a larger percentage gap at
BB BB H
N M # UBr% gaprnodes time # opt % gaprnodes time # opt
30 4 32 1.7 5.5 17.5 40 3.2 3.1 10.7 40
8 24 3.3 42.5 131.9 39 1.7 38.7 124.3 39
40 4 28 1.2 5.7 31.1 40 5.4 2.3 15.0 40
8 25 1.8 23.5 146.6 39 0.8 18.2 117.4 39
50 4 31 0.6 2.7 25.4 40 5.2 1.2 13.5 40
8 24 1.3 11.9 117.3 38 0.5 15.5 144.0 39
60 4 27 0.5 1.8 23.7 40 5.2 1.0 16.2 40
8 23 1.2 12.6 180.6 40 0.4 2.2 37.5 40
Table 2: Performance of different variants of the algorithm in solving the RCB
problem.
the root node as this figure is computed over the entire set of instances (as
opposed to a subset of them for the other algorithm).
4.2.4 Problem variants
We also tested algorithm BB H on some variants of the problem, obtained
combining the following features:
no loans/binary loans/continuous loans: meaning that loans are not available,
can be used only at their maximum amount, or can be used at any intermediate
value;
independent/dependent risk factors: meaning that risk factors may impact the
profit of each project iaccording to a linear or to a non-linear function.
We consider two types of functions (∆i)i∈N : (Q) convex quadratic functions
∆i(ξ
ξ
ξ) = ξ
ξ
ξTQ
Q
Qiξ
ξ
ξ/2 + g
g
giTξ
ξ
ξ, and (L) linear functions ∆i(ξ
ξ
ξ) = g
g
giTξ
ξ
ξ. In the linear
case, the g
g
gicoefficients are generated as in the quadratic case.
Table 3reports the results of our experiments on the six variants described
above. The upper part of the table refers to instances with quadratic risk
functions, while the lower part addresses the linear case.
The results show that, in the quadratic case, the “no loans” and “binary
loans” variants tend to be slightly easier than the “continuous loans” in terms
of number of optimal solutions and average computing time.
As to the linear case, it looks much more challenging that its quadratic
counterpart. This counter-intuitive behaviour is due to the way the instances
are defined, in particular for what concerns functions ∆i(ξ
ξ
ξ). Indeed, the impact
of risk factors in the profit reduction is always larger in the linear case than in
the quadratic case; since we are considering a robust (i.e., worst-case) setting,
this makes the linear case farther from nominal values than the quadratic
one. Nevertheless, also in this challenging case, algorithm BB H is able to solve
almost 75% of the instances with an average time below 10 minutes.
No loans Binary loans Continuous loans
N M time # opt time # opt time # opt
Q 30 4 5.5 40 6.0 40 10.7 40
8 118.4 40 135.8 40 124.3 39
40 4 8.9 40 9.4 40 15.0 40
8 169.0 40 97.2 39 117.4 39
50 4 10.1 40 9.8 40 13.5 40
8 179.5 40 190.2 40 144.0 39
60 4 13.8 40 13.2 40 16.2 40
8 25.1 40 25.7 40 37.5 40
L 30 4 241.1 37 110.7 36 217.3 37
8 580.0 32 618.9 32 480.9 31
40 4 223.8 37 195.6 38 193.1 36
8 447.1 28 276.3 28 247.9 28
50 4 744.8 22 740.1 24 798.6 25
8 523.2 39 238.3 40 251.9 40
60 4 832.9 7 957.4 7 1238.7 8
8 946.1 32 695.0 33 665.5 32
Table 3: Performance of the full algorithm in solving different variants of the
RCB problem.
5 Conclusion
In this work, we studied optimization problems where part of the cost param-
eters are not known at decision time, and the decision flow is modeled as a
two-stage process. In particular, we addressed general problems in which all
constraints (including those linking the first and the second stages) are defined
by convex functions and involve mixed-integer variables. To the best of our
knowledge, this is the first attempt to extend the existing literature to tackle
this wide class of problems.
To this aim, we derive a relaxation of the problem which can be formulated
as a convex optimization problem, and embed it within a branch-and-bound
algorithm where branching occurs on integer and continuous variables. By
combining enumeration and on-the-fly generation of the variables, we obtain
a branch-and-price scheme, for which we prove convergence to ε-optimality.
In addition to the theoretical analysis, we applied our method to two opti-
mization problems affected by objective uncertainty, namely a variant of the
Capacitated Facility Location problem and a capital budgeting problem. Our
computational experiments showed that the proposed method is able to solve
relevant-size instances for both problems.
Funding
This research was supported by “Mixed-Integer Non Linear Optimisation:
Algorithms and Application” consortium, which has received funding from the
European Union’s EU Framework Programme for Research and Innovation
Horizon 2020 under the Marie Sk lodowska-Curie Actions Grant Agreement No
764759.
Appendix A Recalls of convex and concave
conjugate
In this appendix we review some basic results on conjugate functions and
Fenchel duality. For a detailed treatment we refer to Rockafellar (1970).
Let f:Rn→Rbe a given function, its convex conjugate is denoted by
f∗:Rn→Rand is given by
f∗(π
π
π) = sup
x
x
x∈dom(f)π
π
πTx
x
x−f(x
x
x)
Similarly, we denote by g∗the concave conjugate of a given function g:Rn→
R, given by
g∗(π
π
π) = inf
x
x
x∈dom(g)π
π
πTx
x
x−g
g
g(x
x
x)
Note that, if fis a proper convex function and ga proper concave function,
we have that f∗∗ =fand g∗∗ =g. We now state the following Fenchel duality
theorem.
Theorem 1 Let f:Rn→Rbe a proper convex function and g:Rn→Rbe a
proper concave function, then
inf
x
x
x∈dom(f)∩dom(g){f(x
x
x)−g(x
x
x)}= sup
π
π
π∈dom(f∗)∩dom(g∗)g∗(π
π
π)−f∗(π
π
π)
or equivalently,
sup
x
x
x∈dom(f)∩dom(g)
{g(x
x
x)−f(x
x
x)}= inf
π
π
π∈dom(g∗)∩dom(f∗)f∗(π
π
π)−g∗(π
π
π)
Corollary 3 (Maximizing a concave function over a convex set) Let X ⊆ Rnbe a
non-empty convex set, g:Rn→Rbe a proper concave function, then
sup
x
x
x∈X
g(x
x
x) = inf
π
π
πδ∗(π
π
π|X )−g∗(π
π
π)
where δ(x
x
x|X ) = (0 if x
x
x∈ X
+∞otherwise.
Proof The result holds from the fact that sup{g(x
x
x) : x
x
x∈ X } = sup{g(x
x
x)−δ(x
x
x|X )}
and by application of Fenchel duality. More precisely, δ(x
x
x|X ) is convex and, by non-
emptiness of X, is proper.
Notice that Fenchel duality allows the reformulation of an optimization
problem which consists in maximizing a concave function over a convex set
as an unconstrained convex problem since δ∗(·|X) and (−g∗)(·) are convex
functions and positively weighted sums preserve convexity.
h(x
x
x)h∗(π
π
π)
Separable sums
h(x
x
x1, x
x
x2) = f1(x
x
x1) + f2(x
x
x2)h∗(π
π
π1, π
π
π2) = f∗
1(π
π
π1) + f∗
2(π
π
π2)
Scalar multiplications (α > 0)
h(x
x
x) = αf(x
x
x)h∗(π
π
π) = αf∗(π
π
π/α)
Affine mapping composition (det A
A
A6= 0)
h(x
x
x) = f(A
A
Ax
x
x+b
b
b)h∗(π
π
π) = f∗(A
A
A−Tπ
π
π)−b
b
bTA
A
A−Tπ
π
π
Sum with affine functions
h(x
x
x) = f(x
x
x) + a
a
aTx
x
x+b
b
b h∗(π
π
π) = f∗(π
π
π−a
a
a)−b
b
b
Sum of functions
h(x
x
x) =
m
X
i=1
fi(x
x
x)h∗(π
π
π) = inf
v
v
vi,i=1,...,m (m
X
i=1
f∗
i(v
v
vi)
m
X
i=1
v
v
vi=π
π
π)
Table A1: Some convex conjugate calculus rules
Proposition 5 Let fbe a convex function, we have (−f)∗(π
π
π) = −f∗(−π
π
π).
Proof
(−f)∗(π
π
π) = inf
x
x
xnπ
π
πTx
x
x−(−f)(x
x
x)o=−sup
x
x
xn−π
π
πTx
x
x−f(x
x
x)o=−f∗(−π
π
π)
Proposition 6 Let Cbe a convex set such that x
x
x∈C⇔ −x
x
x∈C, we have δ∗(π
π
π|C) =
δ∗(−π
π
π|C)
Proof We have δ∗(π
π
π|C) = supx
x
x∈Cπ
π
πTx
x
x. Denoting x
x
x∗the optimal primal solution
to this optimization problem, there exists u
u
u∈Csuch that u
u
u=−x
x
x∗and thus
δ∗(π
π
π|C) = supu
u
u∈C−π
π
πTu
u
u=δ∗(−π
π
π|C).
Table A1 reports some calculus rules regarding convex conjugates. The
extension to concave conjugates is straightforward.
Appendix B Robust counterpart of
conic-representable uncertainty
sets
We first start by recalling the following strong duality theorem for conic
optimization problems.
Theorem 2 (Conic duality) If the following conic problem has a strictly feasible
solution,
max nc
c
cTξ
ξ
ξ:b
b
b−B
B
Bξ
ξ
ξ∈Ko
then it is equivalent to
min nb
b
bTλ
λ
λ:B
B
BTλ
λ
λ=c
c
c, λ
λ
λ∈K∗o
In the following example, we derive the robust counterpart for general
ellipsoidal uncertainty sets.
Example 6 (Ellipsoidal uncertainty set) We consider the following ellipsoidal uncer-
tainty set:
Ξ = nξ
ξ
ξ: (ξ
ξ
ξ−¯
ξ
ξ
ξ)TP
P
P(ξ
ξ
ξ−¯
ξ
ξ
ξ)≤κ2o
where P
P
Pis a definite positive matrix. We consider the following robust counterpart:
maximize c
c
cTξ
ξ
ξ(B1)
subject to ξ
ξ
ξ∈Ξ (B2)
Let F
F
Fbe a matrix such that P
P
P=F
F
FTF
F
F, we have
(ξ
ξ
ξ−¯
ξ
ξ
ξ)TP
P
P(ξ
ξ
ξ−¯
ξ
ξ
ξ)≤κ2⇔F
F
F(ξ
ξ
ξ−¯
ξ
ξ
ξ)2≤κ(B3)
⇔(κ, F
F
F(ξ
ξ
ξ−¯
ξ
ξ
ξ)) ∈ Qn+1 (B4)
⇔κ
−F
F
F¯
ξ
ξ
ξ−0
−F
F
Fξ
ξ
ξ∈ Qn+1 (B5)
Applying the strong duality theorem, we obtain an equivalent minimization problem:
minimize κµ −(F
F
F¯
ξ
ξ
ξ)Tλ
λ
λ(B6)
subject to −F
F
FTλ
λ
λ=c
c
c(B7)
(µ, λ
λ
λ)∈ Qn+1 (B8)
By inspection, we have λ
λ
λ=−F
F
F−Tc
c
cand thus, one obtains
minimize κµ +¯
ξ
ξ
ξF
F
FTF
F
F−Tc
c
c
|{z }
=c
c
cT¯
ξ
ξ
ξ
(B9)
subject to (µ, −F
F
F−Tc
c
c)∈ Qn+1 (B10)
Note that if P
P
Pis diagonal, i.e., P
P
P= diag(p2
1,...,p2
n), we have F
F
F= diag(p1,...,pn)
and thus F
F
F−T= diag(1/p1,...,1/pn). The resulting problem is therefore
κv
u
u
t
n
X
i=1
c2
i
p2
i
+
n
X
i=1
ci¯
ξi(B11)
Appendix C Generating quadratic functions
for RCB-C-Q
In this appendix, we show how one can generate quadratic functions for the
two-stage robust capital budgeting problem. We therefore consider an instance
Ω for the RCB-C-Q. First, we generate linear functions (∆L
i)i∈N in the same
way as what was done for linear instances. Then, the goal becomes the one of
finding a quadratic functions interpolating functions (∆L
i)i∈N at the extreme