Convexity of elliptically distributed dependent chance constraints

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Convexity of elliptically distributed dependent chance constraints

Hoang Nam Nguyen, Abdel Lisser

To cite this version:

Hoang Nam Nguyen, Abdel Lisser. Convexity of elliptically distributed dependent chance constraints.

2021. �hal-03269257�

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Convexity of elliptically distributed dependent chance constraints

Hoang Nam NGUYEN · Abdel LISSER

Received: date / Accepted: date

Abstract In this paper, we study the problem of linear optimization with probabilistic constraints. We suppose that the constraint row vectors are elliptically distributed. Further, the dependence of the rows is modeled by a family of Archimedean copulas, namely, the Gumbel- Hougaard family of copulas. Under mild assumptions, we prove the eventual convexity of the feasibility set.

Keywords Chance constraints·Archimedean copulas·Elliptical distributions·Convex optimization

Mathematics Subject Classification (2000) MSC 90C15·90C25·90C59

1 Introduction

1.1Chance constrained optimization

We consider the following linear optimization with joint chance constraints min c^Tx

subject to P{V x≤D} ≥p

x∈Q. (1)

where Q is a closed convex subset of Rⁿ ; c ∈ Rⁿ, D := (D₁, ...,DK) ∈ R^K is a deterministic vector,V:=[v₁, ...,vK]^T is a randomk×n- matrix wherevkis a random vector inRⁿ, ∀k=1,2, ...,Kandp∈ [0,1]. We denoteFeasi(Q)the feasibility set of (1).

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for- profit sectors.

Corresponding author : Hoang Nam NGUYEN.

Hoang Nam NGUYEN·Abdel LISSER

Laboratory of signals and systems, Centrale Supélec, 91405 Orsay, France E-mail: [email protected] , [email protected]

Chance-constrained optimization has been widely studied in the literature and plays an important role in engineering, telecommunication, etc. Chance-constrained problem was first introduced by Charnes et al. [11], they studied the individual chance constraint case.

Miller and Wagner [57] dealt with joint chance constraints but only in the independent case. Prekopa [66] was the first researcher who studied joint chance constraints with random parameters on the right hand side. Joint chance-constrained problems are generally non- convex problems. Therefore, several approximations have been proposed in the literature, see for instance [15, 46, 53]. Cheng and Lisser [15] proposed piecewise tangent approximations and sequential approximations to come up with a lower and upper bounds. Luedtke and Ahmed [53] reformulated the original problem by using a mixed-integer linear optimization in finite support case. An efficient way to solve the chance constraints problem is given by the scenario approach studied in [7, 8, 17, 20, 35, 52, 63, 64]. Applications of chance-constrained optimization problems can be found in [9, 27, 28, 37, 39, 43, 48, 72, 74].

1.2Convexity of the feasibility set

The convexity of chance constraints as well as the regularity properties of probability func- tions are difficult issues both from theoretical and computational perspectives. Prekopa [67]

introduced the notion of logarithmic concavity to deal with dependent constraints case.

Borell [7] generalized this notion to r-concavity. Prekopa et al. [69] derive a fundamental result about the convexity of the considered problem if we assume that the rows are normally distributed. Sen [71] introduced a relaxation method for chance-constrained programs with a discrete random variable. Marti [54] studied the differentiation of probability functions by an integral transformation method. The derivatives of the probability function can be obtained by applying an integral transformation to its integral representation. Some basic results on the differentiability of a probability function were studied by Kibzun et al [40].

They proposed new formulations of the gradient of probability functions in different forms.

Lobo [49] studied some applications of second-order cone program leading to a new approach for solving chance constraints. A more developed direction was initialized by Henrion [32]

which gave a full description of the structure (not only the convexity) of a one-row linear optimization with a chance constraint by introducing a new notion of r-decreasing function.

Henrion [33] studied the convexity in the case where the constraints are independent. To deal with the dependent case, Henrion and Strugarek [34], Van Ackooij [77] and Cheng et al. [14] used the theory of copulas to model the dependence of constraints. They supposed that the distribution of the constraint row vectors are elliptically distributed. Under high probability thresholdp, they prove the convexity ofFeasi(Q). Hong et al [35] proposed to solve joint chance-constrained programs by sequential convex approximations. They prove that the solutions of the sequence of approximations converge to a Karush-Kuhn-Tacker (KTT) point of the original problem. Farshbaf-Shaker et al [23] proved some properties of chance constraints in infinite dimensions. They supposed that the feasibility set belongs to a Banach space. Under mild conditions, they proved regularity properties of the probability function with an application to PDE constrained optimization. Wim van Ackooij [79] studied the convexity of the feasibility set in a general framework by using the radial representation of elliptical distributions.

In this paper, we deal with the convexity ofFeasi(Q). Firstly, we suppose that the vectors vi are elliptical distributions. Under mild conditions, we generalize the convexity result in Cheng et al. [14].

This paper is organized as follows. In Section 2, we reformulate the probability function of problem (1). Section 3 studies sufficient conditions for the convexity of Feasi(Q). In Section 4, we show the dependence structure of the constraints of problem (1) and prove the convexity ofFeasi(Q).

2 Elliptical constraints in linear optimization

2.1Preliminaries

Firstly, we recall some important definitions and propositions we use in our paper.

Proposition 1 (Theorem 2.1, [21])LetXa random variable inRⁿbe a spherical distribu- tion. Hence, there exists a functionψ:R→Rsuch that for allt∈Rⁿ, we have

φ_X(t):=E(e^itTX)=ψ(t^Tt)=ψ(t²

1+...+t_n²).

The functionψis called a characteristic generator ofX.

Definition 1 (Definition 2.2, [21])A random variableUinRⁿis an elliptical distribution if we have the following representation

U=µ+AX,

where Xfollows a spherical distribution with a characteristic generator φ,A∈ R^n×n, AA^T =Σandµ∈Rⁿ.

Definition 2 (Chapter 4.6, [68])A function f :R^s→ (0,+∞)is r-concave for a given r∈ (−∞,+∞)ifdomf is convex and

f(αx+(1−α)y) ≥ [αf(x)^r+(1−α)f(y)^r]^1r if r,0. f(αx+(1−α)y) ≥ f(x)^αf(y)^1−α if r=0.

for all x,y ∈dom f andα ∈ [0,1]. In the case wherer=0, f is called a log-concave function. Here,dom f denotes the defined domain of f.

Definition 3 (Definition 2.2, [32])

A function f :R→Ris said r-decreasing for somer∈Rif it is continuous on(0,∞) and if there exists somet^∗ > 0 such that the functiont^rf(t)is strictly decreasing for all t>t^∗.

Definition 4 (Definition 1, [70])Acopulais the distribution functionC:[0,1]^K → [0,1] of a random vector in dimensionKwhose marginals are uniformly distributed on[0,1]. Proposition 2 (Theorem of Sklar, [70])Given a distribution function F : R^K → [0,1] with marginalsF₁, ...,FK, there exists a copulaCsuch that

∀z∈R^K, F(z)=C(F₁(z₁), ...,F_K(z_K)).

Moreover, ifFiis continuous, thenCis uniquely given by

C(u)=F

F⁽⁻¹⁾

1 (u₁), ...,F_K⁽⁻¹⁾(u_K) . In other words,Cis uniquely determined onF₁×...×FK.

Proposition 3 (Fréchet-Hoeffding upper bound)For any copulaCandu=(u₁, ...,u_K) ∈ [0,1]^K, we have

C(u) ≤CM(u):= min

k=1,...,Kuk.

Definition 5 (Definition 2.2, [56])A copulaCis strictlyArchimedeanif there exists a contin- uous and strictly decreasing functionΨ:(0,1] → [0,∞)such thatΨ(1)=0,limt→0Ψ(t)=

∞and

C(u)=Ψ⁽⁻¹⁾

K

Õ

i=1

Ψ(u_i)

! . Ψis called the generator ofC.

Definition 6 (Definition 2.3, [56])A real function f :R→RisK-monotonicon an open intervalI⊆RwithK ≥2 if it is differentiable up to(K−2)- order and the derivatives are satisfied by

(−1)^k d^k

dt^k f(t) ≥0, 0≤k ≤K−2 and∀t∈I.

and the function(−1)^{K−2 dK−2}

dt^K−2f(t)is non increasing and convex onI. Moreover, f is calledcompletely monotonicif f isK−monotonic, ∀K ∈N.

Proposition 4 (Theorem 2.2, [56])Givenψ:(0,1] → [0,+∞)a strictly decreasing function such thatψ(1)=0. Hence, it is the generator of a strictly Archimedean copula in dimension Kif and only ifψ⁻¹is K-monotonic on(0,∞).

2.2Reformulation of the probability function

In problem (1), we suppose thatvifollows an elliptical distribution such that

vi=µ_i+A_iY_i, i=1,2, ...,K (2)

where µ_i ∈ Rⁿ, A_i ∈R^n×n such thatA_iA^T_i = Σ_i is a positive definite matrix,Y_i is a spherical distribution with a characteristic generatorψ_i. We refer the readers to [21] for more details about elliptical distributions.

Assumption 1 0<Q.

In this paper, unless otherwise specified, we assume that Assumption 1 holds.

Let

ξ_i(x):= v^T_i x−µ^T_i x px^TΣ_ix

.

gi(x):= D_i−µ^T_i x px^TΣ_ix

. (3)

The constraint in (1) can be rewritten as follows P{V x≤D} ≥p.

⇔P

v^T_i x≤D_i, i=1, ...,K ≥p.

⇔P{ξ_i(x) ≤gi(x), i=1, ...,K} ≥p. (4) The characteristic function ofξ_i(x)is given by

E(e^itξi(x))=ψ_i(t²) ∀t∈R.

We deduce thatξ_i(x)is a 1-dimension distribution that does not depend onx. Our aim is to reformulate this function to study the convexity ofFeasi(Q). For this purpose, we use the Sklar Theorem (cf. Proposition 2) to rewrite constraint (4) as follows

P{ξi(x) ≤gi(x), i=1, ...,K} ≥p.

⇔C_x[F₁(g₁(x)), ...,F_K(gK(x))] ≥p. (5) whereC_x denotes the Copula of the multivariate variableξ(x):=(ξ₁(x), ..., ξ_K(x))and F_idenotes the cumulative distribution function ofξ_i(x), i=1, ...,K.

We suppose a specific form ofξ(x)which shows explicitly the variation ofξwith respect tox.

In particular, we suppose that for allx, C_xis a strictly Archimedean copula with generator ψ_x. We use an Archimedean copula to model the dependence as we can calculate explicitly its partial derivatives. The constraint (5) can be written as

C_x[F₁(g₁(x)), ...,F_K(gK(x))] ≥p⇔ψ⁽⁻_x¹⁾

K

Õ

i=1

ψ_x(F_i(gi(x)))

!

≥p.

⇔

K

Õ

i=1

ψx(F_i(gi(x))) ≤ψx(p). (6)

We summarize some selected strictly Archimedean copulas in the following table:

Now, we add auxiliary variables{α_i ≥0, i=1, ...,K}in order to reformulate constraint (6) into separated constraints. In particular, asψ_x is positive, constraint (6) is equivalent to the following constraint

Type of copula Parameterθ GeneratorΨθ(t)

Independent - -log(t)

Gumbel-Hougaard θ≥1 [−log(t)]^θ

Frank θ >0 −log

e^−θt−1 e^−θ−1

Clayton θ >0 ¹_θ(t^θ−1) Joe θ≥1 −log[1− (1−t)^θ]

Table 1: Different types of strictly Archimedean copulas.













ψ_x(F_i(gi(x))) ≤α_iψ_x(p), i=1, ...,K. αi ≥0, i=1, ...,K.

ÍK

i=1α_i=1.

(7)

This means that if x^∗ ∈ Feasi(Q)then there existsα^∗ = (α^∗

1, ..., α^∗_K) ∈ R^K such that (x^∗, α^∗) satisfies constraints (7). On the other hand, if(x^∗, α^∗)satisfies constraints (7) and x^∗ ∈ Q, we deduce that x^∗ ∈ Feasi(Q). Moreover, for x^∗ ∈ Feasi(Q), we can choose α^∗=(α^∗

1, ..., α^∗_K)in order to satisfy constraints (7), i.e. , α_i^∗= ψx^∗(F_i(gi(x^∗)))

ÍK

j=1ψ_x^∗(F_j(gj(x^∗))), ∀i=1,2, ...,K. (8) By applying the decreasing monotonicity of the generator ψ_x, constraints (7) can be written as follows













Fi(gi(x)) ≥ψ_x⁽⁻¹⁾(αiψx(p)), i=1, ...,K.

α_i ≥0, i=1, ...,K. ÍK

i=1αi =1.

(9) Remark 1 Our aim is to show the concavity ofF_i(gi)with respect toxand the joint convexity ofψ_x⁽⁻¹⁾(α_iψ_x(p))with respect to(α,x).

2.3Concavity ofF_i(gi)

In this section, we give appropriate sufficient conditions for the concavity ofF_i(gi). Lemma 1 (Lemma 2, section 3.3, Cheng et al. [14])

Letri >1,for some1≤i ≤K. Thengiis(−r_i)−concave on any convex subset of the following subset ofQ

Ω⁽ⁱ⁾:=

x∈Q |D_i−µ^T_i x> ri+1 r_i−1

λ⁻_i,min¹² ||µ_i||p x^TΣ_ix

(10) whereλi,minis the smallest eigenvalue ofΣi.

Remark 2 It is easy to see thatΩⁱis a convex set because^r_rⁱ_i⁺¹₋₁λ_i,min⁻¹² ||µi||>0, µ^T_i xis linear and

px^TΣ_ixis convex with respect tox.

Lemma 2 (Lemma 3, section 3.3, Cheng et al. [14])

IfDi > 0and µi = 0, for some1 ≤i ≤ K, thengiis(−1)−concave on any convex subset of the setQ.

Based on these two Lemmas, we have

Lemma 3 Let I ⊂ {1,2, ...,K}such that µ_i , 0,∀i ∈ I and µ_i = 0, ∀i < I. Letr = (r₁, ...,r_K)such thatr_i >1, ∀i∈Iandr_i=1otherwise. Let

p^∗:=max 1

2 ,max

i∈I

F_i

ri+1 r_i−1

λ⁻_i,min¹² ||µ_i||

. Hence,∀p>p^∗, we have

Conv(Feasi(Q)) ⊂Ù

i∈I

Ωⁱ. Moreover,gi >0and(−r_i)−concave on any convex subset ofÑ

i∈IΩⁱ, i=1,...,K, where Convis the convex hull.

Proof Firstly, we prove that∀p>p^∗,Feasi(Q) ⊂Ñ

i∈IΩⁱ.

In fact, let x₀ ∈ Feasi(Q). In Section 2.2, we show that x₀ satisfies constraint (5). In particular, we have

Cx₀[F₁(g₁(x₀)), ...,FK(gK(x₀))] ≥p. (11) By applying Proposition 3, we deduce that

1) Fi(gi(x₀)) ≥p, ∀i∈I⇒Fi(gi(x₀))>p^∗≥Fi

ri+1

ri−1λ⁻_i,min¹² ||µi||

, ∀i∈I.

AsF_iis increasing monotonic, we have gi(x₀)> r_i+1 ri−1

λ⁻_i,min¹² ||µi||, ∀i∈I.

Therefore,

D_i−µ^T_i x₀> r_i+1 ri−1

λ⁻_i,min¹² ||µ_i||

q x^T

0Σ_ix₀, ∀i∈I.

2) Fj(gj(x₀)) ≥p, ∀j<I⇒Fj(gj(x₀))>p^∗≥ ¹

2, j<I.

AsFj(0)= ¹₂, we have

gj(x₀)>0, ∀j<I.

Therefore,

D_j >0, ∀j<I.

Hence, we deduce thatx₀∈Ñ

i∈IΩⁱ. In other words, we have Feasi(Q) ⊂Ù

i∈I

Ωⁱ. By Remark 2, Ñ

i∈IΩⁱis a convex set. Hence, we have Conv(Feasi(Q)) ⊂Ù

i∈I

Ωⁱ. which ends the first part of Lemma 3.

Secondly, we prove thatgi>0 and(−r_i)−concave on any convex subset ofÑ

i∈IΩⁱ. In fact, this is a direct consequence of Lemma 1 and Lemma 2.

u t

Lemma 4 (Lemma 3.1, Henrion, R. and Strugarek, C. [32])

Let F : R → [0,1]be a distribution function with(r+1)−decreasing density f for somer>0. Hence, the functionz7→F(z^−1r)is concave on(0,(t^∗)^−r), wheret^∗is defined in Definition 3. Moreover,F(t)<1, ∀t∈R.

We use Lemmas 3 and 4 to prove the following auxiliary result

Lemma 5 LetI, µ=(µ₁, ..., µ_K), r=(r₁, ...,r_K)as defined in Lemma 3. We suppose that the cumulative distribution functionF_ihas(r_i+1)−decreasing densities with the thresholds t_i^∗(r_i+1), ∀i=1, ...,Kandp>p^∗, where

p^∗=max 1

2,max

i∈I Fi

ri+1

r_i−1λ⁻_i,min¹² ||µi||

, max

j=1,...,KFj[t^∗_j(r_j+1)]

. Hence, for ally,z∈Feasi(Q)and0≤a≤1, we have

F_i(gi(ay+(1−a)z)) ≥aF_i(gi(y))+(1−a)F_i(gi(z)).

Proof By Lemma 3, we deduce thatgi is(−r_i)−concave andgi >0 onConv(Feasi(Q)), for alli=1, ...,K. Hence, for anya∈ [0,1]andy,z∈Feasi(Q), we have

gi(ay+(1−a)z) ≥ [ag_i^−ri(y)+(1−a)g^−r_i ⁱ(z)]^−ri1. (12) Asy ∈Feasi(Q)andp>p^∗, we have

Cy[F₁(g₁(y)), ...,FK(gK(y))]>p^∗. (13) By Proposition 3 and the definition ofp^∗, we deduce that

F_i(gi(y))>p^∗≥F_i[t_j^∗(r_i+1)], i=1, ...,K. (14)

AsFiare increasing monotonic, we deduce that

gi(y)>t^∗_i(r_i+1)>0⇒0<gi(y)^−ri <(t_i^∗(r_i+1))^−ri, i=1, ...,K. (15) Similarly, we obtain the inequality forz.

By takingFion both sides of (12), we have

F_i(gi(ay+(1−a)z)) ≥F_i([ag^−r_i ⁱ(y)+(1−a)g_i^−ri(z)]^−ri1). (16) By (15) and Lemma 4, we have

F_i(gi(ay+(1−a)z)) ≥F_i([ag^−r_i ⁱ(y)+(1−a)g_i^−ri(z)]^−ri1) ≥aF_i(gi(y))+(1−a)F_i(gi(z)).

u t

In the rest of the paper, we suppose that the statements in Lemma 5 hold.

3 Convexity ofψ⁽⁻¹⁾_x (α_iψ_x(p))

The aim of this section is to study appropriate sufficient conditions for the joint convexity ofU(x, α_i)= ψ⁽⁻_x¹⁾(α_iψ_x(p))with respect to(x, α_i), i=1,...,K. For this purpose, we study the positive semidefiniteness of the Hessian matrix of ψ⁽⁻_x¹⁾(α_iψ_x(p)) on the convex set Q× [k₁,k₂], for givenk₁,k₂where

0≤k₁<k₂ ≤1. (17)

Assumption 2 We suppose thatψ: (x,t) 7→ψx(t)is a jointly continuously differentiable function with respect to (x,t)up to second-order as well asψ⁽⁻¹⁾. Moreover, ψ⁽⁻¹⁾_x is 4- monotonic function with respect tot, ∀x∈Q.

The following lemma is a reformulation of the positive semidefiniteness of the Hessian matrix ofU.

Lemma 6 Suppose that Assumption 2 holds andp<1. Hence, the positive semidefiniteness of the Hessian matrix of U on the convex set Q× [k₁,k₂] is equivalent to the positive semidefiniteness of the following symmetric matrix

"

d² dx²

d² dα_i² −

d² dxdα_i

d² dxdα_i

T#

o

[U(x, αi)]. (18)

for all(x, α_i)onQ× [k₁,k₂].

Proof ForU(x, αi)=ψ_x⁽⁻¹⁾(αiψx(p)), we have d²

dα²_iU(x, αi)=[ψx(p)]²(ψ⁽⁻¹⁾_x )⁰⁰(αiψx(p)).

Asψ⁽⁻_x ¹⁾is 4-monotonic, we deduce that(ψ_x⁽⁻¹⁾)⁰⁰is decreasing. Note thatψ_x(p)>0 as p< 1. If there existsa ≥ 0 such that(ψ_x⁽⁻¹⁾)⁰⁰(a)= 0, by the decreasing monotonicity of (ψ_x⁽⁻¹⁾)⁰⁰, we deduce that(ψ_x⁽⁻¹⁾)⁰⁰(t)=0 on(a,∞)as well asψ⁽⁻_x¹⁾(t)=c₁t+c₂ on(a,∞) wherec₁,c₂ are given scalars. However,ψ⁽⁻_x¹⁾(t) ∈ (0,1], ∀t ∈ [0,∞]which means that ψ⁽⁻_x ¹⁾(t)=c₂on(a,∞), which contradicts the strict monotonicity ofψ_x⁽⁻¹⁾. Hence, we have (ψ⁽⁻¹⁾_x )⁰⁰(αiψx(p))>0 as well as ^d

2

dα_i²U(x, αi)>0 onQ× [k₁,k₂]. The quadratic form of the Hessian matrix ofUat(x, αi)is (s^T,r)

δ²

δiδjU(x, α_i)

(s,r).

=r² d²

dα²_iU(x, α_i)

! +

"

s^T d² dxdαi

U(x, α_i)+ d²

dxdαi

U(x, α_i) T

s

# r+s^T

d²

dx²U(x, α_i)

s.

∀(s,r) ∈R^N×R, where n δ²

δiδjU(x, α_i) o

is the Hessian matrix ofUat(x, α_i).

If we consider this quadratic form as a second-order polynomial function ofrwith strictly positive second-order coefficient, then the positive semidefiniteness ofUis equivalent to the positivity of its quadratic form for allr∈Rwhich is equivalent to the following inequality

s^T d²

dxdαi

U(x, αi) 2

− d²

dα²_iU(x, αi)

! .s^T

d²

dx²U(x, αi)

s≤0, ∀s∈R^N.

⇔s^T

"

d²

dxdα_iU(x, αi) d²

dxdα_iU(x, αi) T

− d²

dα_i²U(x, αi)d²

dx²U(x, αi)

#

s≤0,∀s∈R^N.

⇔

"

d²

dxdα_iU(x, α_i) d²

dxdα_iU(x, α_i) T

− d²

dα²_iU(x, α_i)d²

dx²U(x, α_i)

#

− is a negative semi definite matrix.

u t

Next, we introduce a particular family of generators which meets Assumption 2.

3.1A particular form of the family of the generatorsψ_x We consider the following assumptions

Assumption 3 p<1.

Assumption 4 Let a function ψ : (Q ⊂ Rⁿ) × (0,1] → [0,∞) such that (x,t) 7→

(−logt)^κ(x)1 , whereκ:Q→ (0,1]is a strictly positive function.

If Assumption 4 holds, we have ψ⁽⁻¹⁾_x (t) = e^−tκ(x). We can check thatψx is a strictly decreasing function andψx(1)=0 for allx∈Qandψsatisfies the conditions in Assumption 2. Then, by Proposition 4, we deduce that for allk ≥ 2, ψ_x is the generator of a strictly Archimedean k-dimension copula, ∀x∈Q.

Remark 3 In fact,∀x∈Q,ψxis the generator of a copula in the family of Gumbel-Hougaard copulas. The condition where 0< κ(x) ≤1 is deduced by the defined domain of Gumbel- Hougaard copulas. We can see later that if we use a function κto model the dependence structure ofψwith respect tox, the positive semidefiniteness of the matrix in Lemma 6 is equivalent to the following inequality

κ(x)⁰⁰ ω(κ(x)⁰)(κ(x)⁰)^T. (19)

whereωis a strictly positive scalar,κ(x)⁰⁰is the Hessian matrix ofκatxandκ(x)⁰is the gradient vector ofκatx.

From now and on, we suppose that Assumptions 3 and 4 hold. If we apply Lemma 6, it is sufficient to study the positive semidefiniteness of the matrix

H(U)(x, α_i)=

"

d² dx²

d² dα_i² −

d² dxdαi

d² dxdαi

T#

o

[U(x, α_i)]. (20)

on the convex setQ× [k₁,k₂].

3.2An equivalent condition of the positive semidefiniteness of H(U)

The main step of this section is to reformulate the positive semidefiniteness ofH(U)by an inequality under the form (19).

If we apply the form ofψin Assumption (4) inU(x, αi)=ψ_x⁽⁻¹⁾(αiψx(p)), we can rewrite U(x, α_i)as follows

U(x, αi)=e⁻

αi(−logp)

κ(x)1 κ(x)

.

=p^αiκ(x). (21)

Now, by (21), we calculate explicitly ^d

2

dx², ^d

2

dα²_i and ^d

2

dxdαi ofU. By a direct calculation, we deduce the first formulation as follows











dαdiU(x, α_i)=log(p)p^αiκ(x)κ(x)α^κ(x)−_i ¹.

d²

dα_i²U(x, αi)=κ(x)log(p)α^κ(x)−2_i p^ακ(x)i [κ(x) −1+κ(x)log(p)α_i^κ(x)]. (22) Remark 4 The first and second derivatives in (22) are deduced by the rule of differentiation for composite functions in the 1-dimension case.

Next, we calculate_dx^d ofU. With the chain rule of differentiation for composite functions in the multi-dimension case, we have

d

dxU(x, α_i)=log(p)p^αiκ(x)log(α_i)α_i^κ(x)κ(x)⁰. (23) We differentiate the two sides of (23) withα_iin order to obtain_dxdα^d2 _i

d² dxdαi

U(x, α_i)=log(p)α_i^κ(x)−1p^αiκ(x)[1+κ(x)log(α_i)+log(p)log(α_i)α_i^κ(x)κ(x)]κ(x)⁰. (24) To calculate ^d

2

dx² ofU, we differentiate both sides of (23) withx d²

dx²U(x, αi)=p^αiκ(x)(logp)(logαi)[κ⁰⁰(x)+(logαi+logαilogp.α_i^κ(x))κ(x)⁰(κ(x))^T].

(25) Remark 5 κ(x)⁰is a column vector andκ(x)⁰⁰is a symmetric matrix. We can write

d

dxU(x, α_i) = logp.logα_i.t₁(x).t₂(x).t₃(x) where t₁(x) = p^αiκ(x), t₂(x) = α^κ(x)_i and t₃(x)= κ(x)⁰. We use the product rule of differentiation and apply again the chain rule of differentiation for composite functions in the multi-dimension case in order to obtain (25).

We combine (22) and (25) in order to obtain ^d

2

dα²_i d² dx²

d² dα²_i

d²

dx² =κ(x)log(p)²α_i^2κ(x)−2log(αi)p^2.ακ(x)i × hκ(x) −1+κ(x)logpα_i^κ(x)i

×

hκ(x)⁰⁰+κ(x)⁰(κ(x)⁰)^T(logα_i+logα_i.logp.α^κ(x)_i ) i. (26)

Moreover, ^d

2

dαidx

d² dαidx

T

is deduced from (24) d²

dα_idx d²

dα_idx T

=log(p)²α_i^2κ(x)−2.p^2.αiκ(x)×

1+κ(x)logαi+logplogαi.α^κ(x)_i κ(x)2

κ(x)⁰(κ(x)⁰)^T. (27) Note that log(p)².α^2κ(x)−_i ².p^2.αiκ(x)is the common factor of (26) and (27). Then, we can reduce this term from both sides of (27) and (26) as follows

M=log(p)⁻².α^−2κ(x)+2_i .p^{−2.ακ(x)i} .

"

d² dα_i²

d² dx² − d²

dα_idx d²

dα_idx T#

=

=h

κ(x) −1+κ(x)logp.α_i^κ(x)i .h

κ(x)⁰⁰+κ(x)⁰(κ(x)⁰)^T(logαi+logαi.logp.α^κ(x)_i )i

×κ(x)logα_i−

1+κ(x)logα_i+logplogα_i.α^κ(x)_i κ(x)₂

κ(x)⁰(κ(x)⁰)^T. (28)

Hence, in order to study the positive semidefiniteness of

d² dα_i²

d² dx²− _dα^d2

idx

d² dαidx

T on Q× [k₁,k₂], we study the positive semidefiniteness ofMwhen(x, α) ∈Q× [k₁,k₂].

We rewriteMas follows

M=Aκ(x)⁰⁰−Bκ(x)⁰(κ(x)⁰)^T. (29) where

















A=κ(x)logαi.h

κ(x) −1+κ(x)logp.α_i^κ(x)i . B=κ(x)log(αi)²(1+logp.α_i^κ(x))h

1−κ(x) −κ(x)logp.α_i^κ(x)i +

1+κ(x).logα_i+logp.logα_i.α^κ(x)_i .κ(x)₂ .

3.3A sufficient condition of the convexity ofψ⁽⁻_x¹⁾(α_iψ_x(p))

In (29), AandBdepend on x. The key idea of this section is to study a lower bound of Aand an upper bound ofB. Particularly, we will study the positive semidefiniteness of the following term

M^∗=A₁κ(x)⁰⁰−B₁κ(x)⁰(κ(x)⁰)^T. (30) whereA≥A₁, B≤B₁andA₁,B₁do not depend onx.

Next, we consider the following assumption.

Assumption 5 1. p≥e⁻¹.

2. 0<c_l ≤κ(x) ≤1, ∀x∈Q. 3. 0<h_l ≤α_i ≤h_u <1.

4. There existsδl andδusuch that 0< δl ≤ ||x|| ≤δu < ∞, ∀x ∈Q, where||.||is the Euclidean norm inRⁿ.

In other words,κ(x)andαiare bounded by strictly positive scalarscl,cu,hlandhuand Qis a bounded domain inRⁿ. The main idea is to find a functionκsuch that if Assumptions 3,4 and 5 hold, ψ_x⁽⁻¹⁾(α_iψ_x(p))is a convex function with respect to(x, α)onQ× [h_l,h_u].

Now, we evaluateAandBin (29) as follows

1. We use the statements 1,2 and 3 in Assumption 5 to deduce that 0 ≤ 1−α^κ(x)_i ≤ 1+logp.α^κ(x)_i ≤1+logp.h_l≤1. Then we combine withc_l≤κ(x) ≤1 and(−logα_i) ≥ (−loghu)to deduce a lower bound ofA

A=κ(x)(−logα_i).h

1−κ(x).(1+logp.α^κ(x)_i ) i

≥c_l(−logh_u)(−logp.h_l). (31)

2. Since 0≤1+logp.α_i^κ(x) ≤1, we apply the Cauchy-Schwarz inequality to deduce that (1+logp.α^κ(x)_i )

h

1−κ(x).(1+logp.α^κ(x)_i ) i

≤ 1

4.κ(x) ≤ 1

4c_l. (32) The following inequalities are deduced by Assumption 5

cl ≤κ(x) ≤1. (33)

−logh_u ≤ −logα_i ≤ −logh_l. (34)

1+logp≤1+logp.α_i^κ(x)≤1+logp.h_l. (35) We apply (32)-(35) in order to get

1+logh_l(1+logp.h_l) ≤1+κ(x).logα_i+logp.logα_i.α_i^κ(x).κ(x)

≤1−cl(−loghu)(1+logp).

Therefore, we have

(1+κ(x).logαi+logp.logαi.α^κ(x)_i .κ(x))²

≤max(|1+logh_l(1+logp.h_l)|,|1−c_l(−logh_u)(1+logp)|)².

We combine withκ(x) ≤1 and−logα_i ≤ −logh_lin order to deduce an upper bound of B

B≤max(|1+logh_l(1+logp.h_l)|,|1−c_l(−logh_u)(1+logp)|)² +(logh_l)². 1

4c_l. (36)

Hence, we deduce that a sufficient condition for the positive semidefiniteness ofMon Q× [h_l,hu]is the positive semidefiniteness of the following term onQ

A₁κ(x)⁰⁰−B₁κ(x)⁰.(κ(x)⁰)^T,

whereB₁=max(|1+logh_l(1+logp.h_l)|,|1−c_l(−logh_u)(1+logp)|)²+(logh_l)²._4c¹

l

andA₁=c_l(−logh_u)(−logp.h_l).

Note that A₁ > 0 as−logp.hl >0 by Assumption 3. LetC = ^B_A¹

1, we deduce that the positive semidefiniteness ofMonQ× [h_l,hu]is equivalent to the positive semidefiniteness of the following term onQ

κ(x)⁰⁰−C.κ(x)⁰.(κ(x)⁰)^T. (37)

3.4Boundedness ofαonFeasi(Q)

The main idea of this Section is to show that for allx ∈Feasi(Q), we can chooseαby (8) andhl,hudefined byDi,Σi,µi,p,clsuch thathl ≤αi≤hu, i=1, ...,K.

In fact, by (8), ifx∈Feasi(Q), we can chooseα_isuch that α_i= ψx[F_i(gi(x))]

ÍK

j=1ψ_x[F_j(gj(x))], ∀i=1,2, ...,K. whereψ_x(t)=(−logt)^κ(x)1 andgi(x)= ^D√^i−µTix

x^TΣ_ix. We use the Cauchy-Schwarz inequality to deduce that

| −µ^T_i x| ≤ ||µ_i||.||x||, ∀i=1, ...,K. (38) From linear algebra, we have

px^TΣ_ix≥p

λ_i,min||x||. (39)

Based on inequality (6), Assumption 3, Lemma 4 and the strictly decreasing monotonicity ofψx, we have

0<

K

Õ

j=1

ψ_x[F_j(gj(x))] ≤ψ_x(p). (40)

We apply (38) - (40) and||x|| ≥δ_lto deduce the following inequality gi(x) ≤ |D_i|

pλi,minδl

+ ||µi||

pλi,min

. AsF_iis increasing monotonic, we have

Fi(gi(x)) ≤Fi

|D_i|

pλ_i,minδ_l + ||µ_i||

pλ_i,min

! . We use the decreasing monotonicity ofψ_xto deduce

ψx(F_i(gi(x))) ≥ψx Fi

|D_i| pλ_i,minδl

+ ||µ_i||

pλ_i,min

! ! . Based on the formulation ofα_iin (8), we have

α_i ≥ ψ_x

F_i

|Di|

√λi,mi nδl

+√^{| |µi| |}

λi,mi n

ψ_x(p) .

Note that 0<−logp≤1 (by Condition 1 in Assumption 5 and Assumption 2). Hence, we have

ψx(p) ≤ (−logp). (41)

The following inequality is deduced by Condition 2 in Assumption 5 and the form ofψx

in Assumption 4

ψx Fi

|D_i| pλ_i,minδl

+ ||µi||

pλ_i,min

! !

≥min













−logFi

|D_i| pλ_i,minδl

+ ||µi||

pλ_i,min

! !

, −logFi

|D_i| pλ_i,minδl

+ ||µi||

pλ_i,min

! ! ¹

cl











=mi. (42)

Let h_l = min

1≤i≤K

mi

−logp

. (43)

Hence, we deduce thatαi ≥hl, ∀i=1,2, ...,Kby (41) and (42).

On the other hand, we have αi =1−

n

Õ

j=1,j,i

αj ≤1− (n−1)h_l.

Let h_u=1− (n−1)h_l. (44)

We conclude that 0<h_l≤α_i ≤h_u <1, ∀i=1,2, ...,K. 4 Convexity results for elliptically distributed chance constraints

In this section, we give an example ofκthat satisfies the matrix inequality (37) onQunder Assumption 5 and we show the dependence structure of the constraints. Our aim is to choose d∈Rand a functionκ^∗that satisfies (37) onQsuch that

0<c_l−d≤κ^∗(x) ≤1−d, ∀x∈Q. (45) and letκ=κ^∗+d.

We have the following formulation d²

dx²log(κ^∗(x))= κ^∗(x)κ^∗(x)⁰⁰−κ^∗(x)⁰(κ^∗(x)⁰)^T

κ^∗(x)² . (46)

Assumption 6 κ^∗(x) ≤ _C¹, ∀x ∈Q.

If Assumption 6 holds, the positive semidefiniteness of ^d

2

dx²log(κ^∗(x)) is a sufficient condition for the positive semidefiniteness of the following term

1

Cκ^∗(x)⁰⁰−κ^∗(x)⁰(κ^∗(x)⁰)^T

κ^∗(x)² . (47)

As κ⁰ = (κ^∗)⁰, κ⁰⁰ = (κ^∗)⁰⁰ and κ^∗(x)² > 0, the positive semidefiniteness of (47) is equivalent to the positive semidefiniteness of the following term

1

Cκ(x)⁰⁰−κ(x)⁰(κ(x)⁰)^T. (48) Hence, if 0 <cl−d ≤ κ^∗(x) ≤ min

1 C,1−d

,∀x ∈Q, we deduce that the positive semidefiniteness of _dx^d22log(κ^∗(x))is a sufficient condition for the matrix inequality (37).

On the other hand, the positive semidefiniteness of _dx^d22log(κ^∗(x))onQis equivalent to the convexity of log(κ^∗)onQ. Hence, log(κ^∗)is a convex function onQ.

Letκ^∗=e^q, we need to find a convex functionq:Q⊂Rⁿ→Rsuch thatqsatisfies the following condition

0<c_l−d≤e^q(x) ≤min 1

C,1−d

.

⇔log(c_l−d) ≤q(x) ≤log

min 1

C,1−d , ∀x ∈Q. (49) Under Assumption 4, we can choose a functionqsuch that

q(x)= 1

L||x||²+z, (50)

whereL>0, z∈Rare real numbers such that

log(c_l−d) ≤ 1

Lδ²_l +z≤ 1

Lδ_u²+z≤log

min 1

C,1−d , (51) We deduce the following lemma

Lemma 7 Letδl, δu,cl,hu,hlsuch that Assumptions 3,4 and 5 hold. LetL>0,d∈R,z∈R are real numbers such that

log(c_l−d) ≤ 1

Lδ_l²+z≤ 1

Lδ²_u+z≤log

min 1

C,1−d . whereCis defined in (37). Letκ:Q→ (0,1]such that

κ(x)=

e^{L1| |x| |2+z}+d

. (52)

Hence,U(x, α_i)= ψ⁽⁻_x¹⁾(α_iψ_x(p)) is a convex function with respect to (x, α_i)on the convex setQ× [h_l,hu].

By using the same notations as (3), the dependence structure of vectorsξi(x)is modeled by an Archimedean copulaCx, where

Cx(t)=ψ_x⁽⁻¹⁾

K

Õ

i=1

ψx(t_i)

!

, t=(t₁, ...,tK), 0<ti ≤1, i=1, ...,K. ψ_x(t_i)=(−logt_i)^κ(x)1 .

For a fixed x₀ ∈ Q, the Archimedean copulaC_x

0 is a Gumbel-Hougaard copula with parameterθ= _κ(x¹

0)and the dependence with respect toxis modeled by functionκ. Remark 6 By using||x||²as the form ofq(x)in (50) to model the dependence structure of vectorsξi(x), we deduce that the dependence structure ofξi(x)is unchanged on the contour lines of function||x||². We can replace||x||²by any convex function to model more general cases.

Our main result is given by the following theorem.

Theorem 1 Consider problem (1). Let h_l,h_u defined by (43) and (44). Suppose that the statements in Lemma 7 hold andFeasi(Q),∅. Hence,Feasi(Q)is a convex set.

Proof In Section 3.4, we show that if Feasi(Q) , ∅, we have 0 < h_l ≤ h_u < 1. Let x₁,x₂∈Feasi(Q)andβ∈ [0,1]. We show thaty:=βx₁+(1−βx₂) ∈Feasi(Q).

In fact, letα¹:=(α¹

1, ..., α¹_K)andα²:=(α²

1, ..., α²_K)given by (8). We deduce that(x¹, α¹) and(x², α²)satisfy constraint (9). In Section 3.4, we show that hl ≤ α_i¹, α_i² ≤ hu, ∀i = 1,2, ...,K. By Lemma 7, we deduce thatU(x, αi)=ψ_x⁽⁻¹⁾(αiψx(p))is a convex function with respect to(x, αi)on the setQ× [h_l,hu].

Hence, we deduce the two following inequalities fory:

F_i(gi(y)) ≥βF_i(gi(x₁))+(1−β)F_i(gi(x₂)).

U(y, βα¹+(1−β)α²) ≤βU(x₁, α¹)+(1−β)U(x₂, α²). (53) By (53), we deduce that(y, βα¹+(1−β)α²)also satisfies (9) as well asy∈Feasi(Q). u t Now, we discuss about Assumption 1. Firstly, we can state thatαicannot go to 0 in (29).

Ifα_i →0, we haveA= O(logα_i)andB =O(logα_i)². Hence, ^B_A → ∞. We deduce that κ(x)⁰=0, ∀x∈Q. Therefore,κis a scalar function and we return to the trivial case where the copulaCxdoes not depend onxwhich is the independent case. Hence,αimust be lower bounded by a strictly positive scalarhl. Based on (8), we deduce thatψx(F_i(gi(x)))is lower bounded by a strictly positive scalar as well asgi(x)is upper bounded. Therefore, ^Di−µ

T ix

√

x^TΣ_ix is upper bounded. Ifx→0, we might have

D_i−µ^T_ix

√

x^TΣ_ix

→ ∞ifDi >0. Hence, we do not have the convexity in this case. We deduce that there exists a strictly positive scalarδlsuch that

||x|| ≥δl, ∀x∈Q.