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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