Shen Peng
email: shenp@kth.se

Francesca Maggioni
email: francesca.maggioni@unibg.it

Abdel Lisser
email: abdel.lisser@l2s.centralesupelec.fr

Bounds for Probabilistic Programming with Application to a Blend Planning Problem

Keywords: stochastic programming, joint chance-constraints, bounds, copulas, blending problem

). The dependent case has been modelled via copulas. New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. When the convexity condition can not hold, an efficient sequential convex approximation approach is further proposed to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. Extensive numerical results on a blend planning problem under uncertainty are finally provided allowing to compare the proposed bounds with the Second Order Cone (SOCP) formulation and Sample Average Approximation (SAA).

Introduction

Chance constrained optimization problems are an important class of optimization problems under uncertainty which involve constraints that are required to hold with specified probabilities. The reader is referred to [START_REF] Prékopa | Probabilistic programming[END_REF][START_REF] Shapiro | Lectures on stochastic programming: modeling and theory[END_REF] for in-deep results and extensive reviews on the theory and applications of chance constrained optimization problems.

Several applications of chance constraints are considered in economics and finance [START_REF] Atwood | Chance-constrained financing as a response to financial risk[END_REF], water reservoir management [START_REF] Andrieu | A model for dynamic chance constraints in hydro power reservoir management[END_REF][START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], system optimization [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], the electrical industry [START_REF] Zorgati | Optimizing financial and physical assets with chance-constrained programming in the electrical industry[END_REF], optimal power flow [START_REF] Zhang | Chance constrained programming for optimal power flow under uncertainty[END_REF] and many others.

The main difficulty of this class of problems is that their feasible sets are generally non-convex. On this purpose [START_REF] Prekopa | On probabilistic constrained programming[END_REF] investigated a wide family of logarithmically concave distributions, showing that under this assumption the feasible set is convex. For the case of symmetric elliptical probability distributions, convex relaxations or reformulations via Second Order Cone Programming (SOCP) were proposed in [START_REF] Henrion | Structural properties of linear probabilistic constraints[END_REF][START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF][START_REF] Cheng | A completely positive representation of 0-1 linear programs with joint probabilistic constraints[END_REF][START_REF] Cheng | Chance constrained 0-1 quadratic programs using copulas[END_REF][START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF].

In order to solve chance constrained problems efficiently, we need both the convexity of the corresponding feasible set and efficient computability of the considered probability [START_REF] Nemirovski | Convex approximations of chance constrained programs[END_REF]. This combination is rare, and very few are the cases in which chance constraints can be processed efficiently (see [START_REF] Dentcheva | Concavity and efficient points of discrete distributions in probabilistic programming[END_REF][START_REF] Lagoa | Probabilistically constrained linear programs and risk-adjusted controller design[END_REF]). Moreover, when the random variables are not elliptically distributed, e.g., truncated distributions, SOCP cannot be used. Whenever this is the case, bounds and tractable approximations of chance constraints can be very useful. On this purpose, [START_REF] Prékopa | Boole-bonferroni inequalities and linear programming[END_REF][START_REF] Prékopa | Sharp bounds on probabilities using linear programming[END_REF] present a method to obtain sharp lower and upper bounds based on the knowledge of some of the binomial moments of the number of events which occur, using linear programming. In particular [START_REF] Prékopa | Boole-bonferroni inequalities and linear programming[END_REF] shows that by the solution of an aggregated LP problem one can get any Bonferroni-type bound in an algorithmic way. Partial disaggregation of the LP problem has been considered by [START_REF] Prékopa | Bounding the probability of the union of events by the use of aggregation and disaggregation in linear programs[END_REF], where the obtained lower bounds generalize the bounds of [START_REF] De Caen | A lower bound on the probability of a union[END_REF] and [START_REF] Kuai | A lower bound on the probability of a finite union of events[END_REF]. Further Bonferroni-type inequalities and a summary of them can be found in the book [START_REF] Galambos | Bonferroni-type inequalities with applications[END_REF] and [START_REF] Bukszár | Computing bounds for the probability of the union of events by different methods[END_REF]. Improved bounds on the probability of the union of events some of whose intersections are emptly are discussed in [START_REF] Yoda | Improved bounds on the probability of the union of events some of whose intersections are empty[END_REF]. [START_REF] Subasi | New bounds for the probability that at least k-out-of-n events occur with unimodal distributions[END_REF] improve the previous bounds using the shape information of the distribution of the random variable based on the knowledge of some binomial moments.

Another class of bounds relies on (hyper)graph structures, see [START_REF] Bukszár | Computing bounds for the probability of the union of events by different methods[END_REF] and references therein.

A computationally tractable approximation of chance constrained prob-lems could also be given by scenario approaches, based on Monte Carlo sampling techniques [START_REF] Luedtke | A sample approximation approach for optimization with probabilistic constraints[END_REF][START_REF] Nemirovski | Convex approximations of chance constrained programs[END_REF][START_REF] Pagnoncelli | Sample average approximation method for chance constrained programming: theory and applications[END_REF], where the probabilistic constraint is replaced by a sampled set of constraints. The sample size is chosen to guarantee that a solution to the sampled problem is feasible to the probabilistic constrained one with a high probability. See [START_REF] Nemirovski | On safe tractable approximations of chance constraints[END_REF] for a survey of safe and scenario approximations of chance constraints. An alternative to scenario approaches consists in providing bounds based on using deterministic analytical approximations of chance constraints. For the case of individual chance constraint, the bounds are mainly based on extensions of Chebyshev inequality which requires the knowledge only of the first two moments of the distribution [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF][START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF]. For joint chance constraints, deterministic equivalent approximations have been discussed in [START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF][START_REF] Cheng | A completely positive representation of 0-1 linear programs with joint probabilistic constraints[END_REF][START_REF] Cheng | Chance constrained 0-1 quadratic programs using copulas[END_REF][START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF] and for special distributions, such as the multivariate gamma, in [START_REF] Szántai | Evaluation of a special multivariate gamma distribution function[END_REF]. In [START_REF] Chen | From cvar to uncertainty set: Implications in joint chance-constrained optimization[END_REF] a new formulation for approximating joint chance constrained problems that improves upon the standard approach using Bonferroni inequality is proposed. The approach decomposes the joint chance constraints into a problem with individual chance constraints, and then applies safe robust optimization approximation on each one of them. Connections with bounds on the conditional-value-at-risk (CVaR) measure are also provided. Besides, in [START_REF] Nemirovski | Convex approximations of chance constrained programs[END_REF] a class of analytical approximations of single and joint independent chance constraints are developed and referred to as Bernstein approximations. Relaxations and approximations of linear chance constraints in the setting of a finite distribution of the stochastic parameters has been studied in [START_REF] Sen | Relaxations for probabilistically constrained programs with discrete random variables[END_REF][START_REF] Prékopa | Dual method for a one-stage stochastic programming problem with random rhs obeying a discrete probability distribution[END_REF][START_REF] Prekopa | Programming under probabilistic constraint with discrete random variable[END_REF][START_REF] Dentcheva | Concavity and efficient points of discrete distributions in probabilistic programming[END_REF][START_REF] Ahmed | Relaxations and approximations of chance constraints under finite distributions[END_REF][START_REF] Dentcheva | On convex probabilistic programming with discrete distributions[END_REF][START_REF] Ruszczyński | Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra[END_REF]. The case of integer programs with probabilistic constraints has been addressed in [START_REF] Dentcheva | Bounds for probabilistic integer programming problems[END_REF] and bounds via binomial moments proposed. They also show how limited information about the distribution can be used to construct such bounds.

Relaxations for probabilistically constrained stochastic programming problems in which the random variables are in the right-hand sides of the stochastic inequalities defining the joint chance constraints are reviewed and provided in [START_REF] Lejeune | Relaxations for probabilistically constrained stochastic programming problems: review and extensions[END_REF].

The convexity of chance constraints with dependent random variables modeled via copulae and random right-hand side has been investigated in [START_REF] Henrion | Convexity of chance constraints with dependent random variables: the use of copulae[END_REF].

In this paper, we study deterministic inner approximations (restrictions) for single and joint independent or dependent probabilistic constraints. The dependent case has been modelled via copulas. The derived upper bounds are based on classical inequalities from probability theory such as the one-sided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see [START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF]). Notice that our proposed bounds do not require any particular assumption on probability distributions. We derive new assumptions under which the bounds based approximations of joint chance constrained problems are convex and the approximated problem can be optimized efficiently. When the convexity condition can not hold, an efficient sequential convex approximation approach is further proposed to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. To the best of our knowledge, these results are new in the literature since the majority of the abovementioned contributions deals with symmetric elliptical distributions.

The bounds are tested on a refinery multiproduct blend planning problem with uncertain raw materials qualities, where each product is required to be on specification with a high probability (see [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF]). The associated optimization model is a joint probabilistic constrained optimization model in which the number of joint independent chance constraints equals the number of final products. In each joint chance constraint the uncertainty appears in the left-hand side coefficient matrix having independent or dependent matrix vector rows, and it ensures that all the qualities are on specification with a high probability. Extensive numerical results on this problem are provided allowing to compare the proposed bounds with the Second Order Cone (SOCP) formulation for individual chance constraints and Sample Average Approximation (SAA) for joint chance constraints.

The main contributions and research questions of the paper can be summarized as follows:

to review inner approximations for individual chance constraints based on classical probabilistic inequalities such as the one-sided Chebyshev, Chernoff, Bernstein and Hoeffding inequalities (see [START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF]); to extend such approximations to the joint independent and dependent case with copulas; to derive new sufficient conditions under which the aforementioned approximations are tractable via logarithmic transformation; to propose a sequential convex approximation method for the cases in which a logarithmic transformation cannot be applied; to provide an extensive numerical campaign based on a blending problem, with the aim of:

understanding the performance of the considered inner approximations in terms of percentage GAP, CPU time and optimal blending recipes with respect to the exact SOCP reformulation for the single chance constraint case and with respect to the Sample Average Approximations (SAA) method for the joint chance constraints case; reducing the computational complexity of the derived problems via a piecewise linear approximation based on tangent and segment approximations; analyzing the sensitivity of the solution for increasing values of considered products qualities; analyzing the performance of the solutions obtained using the aforementioned bounds over the realization of different probability distributions including truncated distributions; providing managerial insights on the usage of bounds.

The paper is organized as follows: Section 1 revises basic facts about copulas necessary for our following investigation. Section 3 investigates how to derive Chebyshev, Chernoff, Bernstein and Hoeffding bounds both for individual and joint probabilistic constrained problems. A refinery blending problem under uncertainty is described in Section 4 and a probabilistic constrained model for this problem presented. Numerical results on the refinery blend planning problem are in Section 5. Conclusions follow.

Basic facts about copulas

We first mention some basic facts about copulas necessary for our following investigation. For most of the notions here we refer to the book [START_REF] Nelsen | An introduction to copulas[END_REF].

Definition 1. A copula is the distribution function C : [0, 1] K → [0, 1]
of some K-dimensional random vector whose marginals are uniformly distributed on [0, 1].

Proposition 1 (Sklar's Theorem [START_REF] Sklar | Fonctions de repartition an dimensions et leurs marges[END_REF]). For any K-dimensional distribution function F : R K → [0, 1] with marginals F 1 , . . . , F K , there exists a copula C such that

∀z ∈ R K , F (z) = C(F 1 (z 1 ), . . . , F K (z K )).
If, moreover, F k are continuous, then C is uniquely given by

C(u) = F (F -1 1 (u 1 ), . . . , F -1 K (u K )).
Otherwise, C is uniquely determined on range F 1 × . . . × range F K .

Sklar's Theorem allow us to handle an arbitrary dependence structure. Notice that if we know the marginal distributions F k together with the copula representing the dependence, we can exactly determine the joint distribution. Secondly, the copula can be uniquely derived from the knowledge of the joint and all marginal distributions.

Consider the following two classes of copulas:

1. Independent (product) copula, defined by

C (u) := K k=1 u k .
Indeed, the independent copula represents the joint distribution of independent random variables u k , k = 1, . . . , K. 2. Gumbel-Hougaard family of copulas, given for a θ ≥ 1 by

C θ (u) := exp    - K k=1 (-ln u k ) θ 1/θ    .
It is easy to see that the independent copula is a special case of the Gumbel-Hougaard copula with θ = 1.

Bounds for probabilistic constrained problems

We consider the following joint chance constrained linear program:

min x c T x s.t. P {Ξx ≤ H} ≥ α, x ∈ X, (1)
where

H = (h 1 , . . . , h K ) ∈ R K , Ξ = [ξ 1 , . . . , ξ K ] T is a K × n random matrix, with ξ k , k = 1, .
. . , K a random vector in R n . We denote with P a probability measure, x a decision vector with feasible set X ⊆ R n + , c ∈ R n and 0 < α < 1 a prespecified confidence parameter. Notice that the objective function parameters c can also be considered as random variables. For the sake of simplicity we replace them by their means. Our goal is to come up with a deterministic equivalent problem of (1) such that the feasible set S(α) := {x ∈ X :

P Ξ T x ≤ H ≥ α} of (1) is convex.
In particular, an individual chance constrained problem can be written as follows:

min x c T x s.t. P ξ T x ≤ h ≥ α, x ∈ X.
(

) 2
If we consider the case of a multivariate normally distributed vectors ξ with mean ξ = E(ξ) and positive definite variance-covariance matrix Σ, the following relations hold true:

P(ξ T x ≤ h) ≥ α, (3)
⇕ ξT x + F -1 (α) Σ 1/2 x ≤ h, (4)
where F -1 (•) is the inverse of F , the standard normal cumulative distribution function. The same scheme can be applied to elliptical distributions, e.g., normal distribution, Laplace distribution, t-Student distribution, Cauchy distribution, Logistic distribution [START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF][START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF].

We now consider the joint chance constrained case. For each k = 1, . . . , K, we assume that ζ k (x) = ξ T k x. Therefore, the chance constraint P{Ξx ≤ H} ≥ α can be equivalently rewritten as

P{ζ k (x) ≤ h k , ∀k} ≥ α.
We have the following result:

Lemma 2. If the random vector (ζ 1 (x), • • • , ζ K (x))
T has a joint distribution driven by the Gumbel-Hougaard copula C θ with some θ ≥ 1, then the chance constraint P{Ξx ≤ H} ≥ α can be equivalently reformulated as

P{ξ T k x ≤ h k } ≥ α y 1/θ k , k = 1, . . . , K, K k=1 y k = 1,
y k ≥ 0, k = 1, . . . , K.
The proof is similar to the proof of Lemma 1 in [START_REF] Cheng | Chance constrained 0-1 quadratic programs using copulas[END_REF].

In particular, if ξ k , k = 1, . . . , K are independent row vectors, P {Ξx ≤ H} ≥ α is equivalent to

K k=1 P ξ T k x ≤ h k ≥ α = K k=1 α y k , (5)
with K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K and y = (y 1 , . . . , y K ) T . Remark 3. We can notice that, when θ = 1, the reformulation with Gumbel-Hougaard copula is equivalent to the reformulation under independent assumption.

When the probability distributions are not elliptical or not known in advance, lower and upper bounds on the individual or joint chance constraints, can be very useful. We will investigate them in the following sections.

Chebyshev and Chernoff Bounds

In the following, we provide bounds for problems (1) and (2) based on deterministic approximations of probabilistic inequalities such as the onesided Chebyshev and Chernoff inequalities.

Chebychev bounds

We consider the one-sided Chebyshev inequality [START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF][START_REF] Lin | Probability inequalities[END_REF]. We assume that ξ has finite second moments and denote σ 2 ξ = V ar(ξ) and ξ = E(ξ) its mean.

The one-sided Chebyshev inequality is given by

P(ξ -ξ ≥ h) ≤ σ 2 ξ σ 2 ξ + h 2 . (6)
For the individual chance constraint problem (2), we have the following result: Proposition 2. Assume that ξ has finite first and second moments with variance-covariance matrix Σ and mean ξ. Under one-sided Chebyshev inequality [START_REF] Cheng | Chance constrained 0-1 quadratic programs using copulas[END_REF], an inner approximation of Problem (2) is obtained as follows min

x c T x s.t. ξT x + α 1 -α Σ 1/2 x ≤ h, x ∈ X. (7)
Moreover, ( 7) is a convex problem.

Proof. First, we note that, assuming P(ξ T x = h) = 0 a.s., then

P ξ T x ≤ h ≥ α, (8)
⇕ P ξ T x≥h ≤ 1 -α, (9) ⇕ P ξ T x -ξT x≥h -ξT x ≤ 1 -α. (10)
Then, we apply (6) to [START_REF] Dentcheva | Concavity and efficient points of discrete distributions in probabilistic programming[END_REF]:

P ξ T x -ξT x≥h -ξT x ≤ σ 2 ξ σ 2 ξ + (h -ξT x) 2 , (11)
where σ 2 ξ = x T Σx with variance-covariance matrix Σ. If

σ 2 ξ σ 2 ξ +(h-ξT x) 2 ≤ 1 -α, then (8)
will be satisfied. Therefore,

x T Σx x T Σx + (h -ξT x) 2 ≤ 1 -α, ⇐⇒ α 1 -α x T Σx ≤ (h -ξT x) 2 ,
which is equivalent to

α 1 -α Σ 1/2 x ≤ h -ξT x. (12)
In the following, we extend our results to the case of joint chance constraints.

We now provide an upper bound to problem (1) based on the one-sided Chebyshev inequality. We assume that ξ k , k = 1, . . . , K has finite second moments. Let σ ξ k = V ar(ξ k ) be the variance of ξ k with variance-covariance matrix Σ k and ξk = E(ξ k ) its mean. The following result holds true: Proposition 3. Assume that the random vectors ξ k , k = 1, . . . , K have a joint distribution driven by the Gumbel-Hougaard copula C θ with some θ ≥ 1. Based on one-sided Chebyshev inequality, an inner approximation of joint chance constrained problem (1) can be obtained by solving the following non linear optimization problem min

x,y c T x s.t. ξT k x + α y 1/θ k 1 -α y 1/θ k Σ 1/2 k x ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X. (13)
Proof. From Lemma 2 and the inner approximation for individual chance constraint based on Chebyshev inequality, we can immediately obtain the conclusion.

Assumption 4. X = R n + ∩ L, L is selected such that Z = {z ∈ R n : z j = ln(x j ), j = 1, . . . , n, x ∈ L} is convex.
Problem [START_REF] Galambos | Bonferroni-type inequalities with applications[END_REF] is not convex but biconvex (see [START_REF] Gorski | Biconvex sets and optimization with biconvex functions: a survey and extensions[END_REF] for the definition) because of the first group of constraints. To come-up with a tractable convex reformulation, with Assumption 4, we use the following logarithmic transformation z = ln x. In this case, Problem (13) can be reformulated as follows

min z,y c T e z s.t. ξT k e z + Σ 1/2 k e ln   α y 1/θ k 1-α y 1/θ k   •en+z ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, z ∈ Z. ( 14
)
where e n is an n × 1 vector of ones. We now prove that problem [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF] is convex for all α ∈ [0, 1]. Lemma 5. Given the sets X, Y, Z with X, Y convex sets; let f : X → Y be a convex function in C 2 and g : Y → Z be a nonincreasing concave function in C 2 . Then, g • f : X → Z is a concave function.

The proof is given in Appendix A. Assumption 6. c ≥ 0. For each k = 1, . . . , K, all the components of ξk and Σ Proof. To show the convexity of problem [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], we firstly need to show the convexity of ln

α y 1/θ k 1-α y 1/θ k in y k , when α y k 1-α y k ≥ 0. As ln α y 1/θ k 1-α y 1/θ k = 1 2 y 1/θ k ln α -ln 1 -α y k 1/θ and y 1/θ k ln α is convex since y 1/θ k
is concave, we can deduce the convexity of function ln

α y k 1/θ 1-α y k 1/θ if the term ln 1 -α y k 1/θ is concave. Since p → ln (1 -p)
is decreasing and concave with respect to p and y k → α y 1/θ k is convex as θ ≥ 1 with respect to y k , we have that ln 1 -α y 1/θ k is concave with respect to y k as shown by Lemma 5.

Since the norm is a convex function and it is also a nondecreasing function on nonnegative space, then the composition function Σ

1/2 k e ln   α y 1/θ k 1-α y 1/θ k   •en+z
is a convex function. Moreover, the function z → ξT k e z in problem ( 14) is convex because ξk ≥ 0. Hence, the problem ( 14) is convex for all α ∈ [0, 1].

When Assumption 4 does not hold, a natural approach is to apply the sequential convex approximation by adjusting y k , k = 1, . . . , K with respect to x. Following [START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF], given

y * k such that K k=1 y * k = 1, y * k ≥ 0, we first fix y k = y * k and obtain x * by solving min x c T x s.t. ξT k x + α (y * k ) 1/θ 1 -α (y * k ) 1/θ Σ 1/2 k x ≤ h k , k = 1, . . . , K, x ∈ X, (15)
and then fix x = x * and update y k by solving min

y ψ T y s.t. α y 1/θ k 1 -α y 1/θ k ≤ h k -ξT k x * Σ 1/2 k x * , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K. (16)
Here, ψ is a chosen searching direction. The constraint

α y k 1/θ 1 -α y k 1/θ ≤ h k -ξT k x * Σ 1/2 k x *
, can be reformulated as

y k ≥ log α (τ n k ) 2 1 + (τ n k ) 2 θ ,
where [START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF] shows the convergence of this algorithm, which provides an upper bound for problem (13).

τ n k = h k -ξT k x * Σ 1/2 k x * . Theorem 2 in

Chernoff bounds

We consider now the Chernoff bound :

P(ξ ≥ h) ≤ E(e tξ ) e th , (17)
where E(e tξ ) is the moment generating function of the random variable ξ and t > 0. We denote with ξ the mean of ξ and with σ 2 = V ar(ξ) its variance.

First, we proof the convexity of E(e tξ T x ).

Lemma 7. For any t > 0, x → E(e tξ T x ) is a convex function.

The proof is given in Appendix A.

Proposition 5. If ξ follows a normal distribution with mean vector ξ and variance-covariance matrix Σ, under Chernoff bound [START_REF] Henrion | Convexity of chance constraints with dependent random variables: the use of copulae[END_REF], an inner approximation of Problem ( 2) is obtained as follows min

x c T x s.t. ξT x + 2 ln 1 (1 -α) Σ 1/2 x ≤ h, x ∈ X. (18)
Moreover, Problem ( 18) is a convex optimization problem.

Proof. First, we have from ( 9)

P(ξ T x ≤ h) ≥ α ⇐⇒ P(ξ T x ≥ h) ≤ 1 -α.
This implies

P(ξ T x ≥ h) ≤ E(e tξ T x ) e th . (19)
Given t > 0, if we choose E(e tξ T x )

e th ≤ 1 -α, then we get an upper bound to problem (2) with feasible region

S(α) = x ∈ X ⊆ R n + |E(e tξ T x ) ≤ (1 -α)e th , (20)
which is convex as x → E(e tξ T x ) is convex, as shown by Lemma 7.

If ξ is a normal distribution with mean ξ and variance-covariance Σ, i.e.

ξ ∼ N ( ξ, Σ) then in [START_REF] Lagoa | Probabilistically constrained linear programs and risk-adjusted controller design[END_REF] we have E(e tξ T x ) = e t ξT x • e 1 2 x T Σxt 2 . The feasible region S(α) can be written as:

S(α) = x ∈ X ⊆ R n + |∃ t > 0 : 1 2 x T Σxt 2 + t ξT x -th ≤ ln(1 -α) . (21)
The set ( 21) is equivalent to:

inf t>0 1 2 x T Σxt 2 + t ξT x -th ≤ ln(1 -α). ( 22
) From d dt ( 1 2 x T Σxt 2 + t ξT x -th) = 0, we get t = h-ξT x x T Σx . Since t > 0 we require h -ξT x > 0. Therefore (21) is equivalent to S(α) = x ∈ X ⊆ R n + | -(h -ξT x) 2 ≤ 2 ln(1 -α)x T Σx , (23)
which is equivalent to the following convex set:

S(α) = x ∈ X ⊆ R n + |h -ξT x ≥ 2 ln 1 (1 -α) ∥Σ 1/2 x∥ . (24)
We extend our results to the case of joint chance constraints.

If we assume that ξ k , k = 1, . . . , K are multivariate normally distributed row vectors having a joint distribution driven by the Gumbel-Hougaard copula, with mean vector ξk = ( ξk1 , . . . , ξkn ) T and covariance matrix Σ k , we can derive a deterministic reformulation of problem (1) based on [START_REF] Chen | From cvar to uncertainty set: Implications in joint chance-constrained optimization[END_REF]. We consider now an upper bound to problem (1) based on Chernoff bound. Proposition 6. If ξ k , k = 1, . . . , K are normally distributed with mean vector ξk and covariance matrix Σ k having a joint distribution driven by the Gumbel-Hougaard copula C θ with some θ ≥ 1, based on Chernoff bound, an inner approximation of Problem ( 1) is obtained by solving the following problem

min z,y c T x s.t. ξT k x + 2 ln 1 1 -α y k 1/θ ∥Σ 1/2 k x∥ ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X. (25)
Proof. First, we note that

P(ξ T k x ≤ h k ) ≥ α y k 1/θ ⇐⇒ P(ξ T k x ≥ h k ) ≤ 1 -α y k 1/θ , k = 1, . . . , K.
Chernoff bound leads to

P(ξ T k x ≥ h k ) ≤ E(e tξ T k x ) e th k , k = 1, . . . , K, (26)
given t > 0. An upper bound to problem (1) is then obtained by solving the following problem for a given t > 0:

min z,y c T x s.t. E(e tξ T k x ) ≤ (1 -α y k 1/θ )e th k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X. (27)
However, if the probability distributions of ξ k , k = 1, . . . , K are not known, the main difficulty of the model ( 27) is given by the computation of E(e tξ T k x ). On the other hand, if we assume ξ k , k = 1, . . . , K are normal distributions with mean ξk and variance-covariance Σ k , i.e.

ξ k ∼ N ( ξk , Σ k ), then we have that E(e tξ T k x ) = e t ξT k x • e 1 2 x T Σ k xt 2 , k = 1, . . . , K. Consequently problem (27) can be written as min z,y c T x s.t. 1 2 x T Σ k xt 2 + t ξT k x -th k ≤ ln(1 -α y k 1/θ ), k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X. (28)
Similarly to the individual chance constraint case, we have:

min z,y c T x s.t. h k -ξT k x ≥ 2 ln 1 1 -α y k 1/θ ∥Σ 1/2 k x∥, k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X.
Problem [START_REF] Maggioni | The value of the right distribution in stochastic programming with application to a newsvendor problem[END_REF] is not a convex optimization problem. Therefore, with Assumption 4, we apply the transformation z = ln x and get:

min z,y c T e z s.t. ξT k e z + Σ 1/2 k e ln    2 ln   1 1-α y 1/θ k     +z ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, z ∈ Z. (29)
Moreover, if ξk ≥ 0, k = 1, 2, . . . , K, and the function ln 2 ln 29) is convex. The following lemma shows the convexity of ln 2 ln

1 1-α y 1/θ k is convex, then Problem (
1 1-α y 1/θ k . Lemma 8. If α ≥ 1-e -1 ≈ 0.6321, then the function y k → ln 2 ln 1 1-α y 1/θ k is convex.
The proof is given in Appendix A. Therefore, when c ≥ 0, α ≥ 1 -e -1 , ξk ≥ 0, k = 1, 2, . . . , K, problem (29) is convex. When Assumption 4 and the convexity condition for problem [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF] do not hold, a sequential convex approximation algorithm can be applied to problem [START_REF] Maggioni | The value of the right distribution in stochastic programming with application to a newsvendor problem[END_REF] in a similar way of what proposed before for Chebyshev bounds.

Notice that SOCP requires the inverse of a CDF which is typically a difficult function to deal with in the case of joint chance constraints. However, Chernoff bound doesn't require any CDF function, and its main advantage is then its easy-use especially from implementation point of view (the CDF is not implemented in current software platform developments). Moreover, the Chernoff bound can be applied with any distribution if the corresponding generated moment function can be reformulated explicitly.

Bernstein and Hoeffding Bounds

Bernstein and Hoeffding bounds are considered as exponential type estimates of probabilities. These inequalities are frequently used for investigating for instance the law of large numbers. They are also often used in statistics and probability theory. In this section, we investigate these bounds for the case of individual (2) and joint chance constraints (1).

Bernstein bounds

In this section, we consider Bernstein bound [START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF]. We assume that the mean and the range parameters for all independent components ξ i of the random vector ξ are known, i.e. l i ≤ ξ i ≤ u i , and E(ξ i ) = ξi , for i = 1, . . . , n.

The Bernstein-type exponential estimate, given by

e -g * h n i=1 u i -ξi u i -l i e g * l i + ξi -l i u i -l i e g * u i ≤ α, (30)
with arbitrary g * > 0, implies P(

n i=0 ξ i ≥ h) ≤ α.
Proposition 7. Given a random vector ξ with components ξ i such that l i ≤ ξ i ≤ u i , and E(ξ i ) = ξi , for i = 1, . . . , n, an upper bound for problem [START_REF] Andrieu | A model for dynamic chance constraints in hydro power reservoir management[END_REF] can be obtained by solving the following problem min

x c T x s.t. n i=1 ln u i -ξi u i -l i e g * l i x i + ξi -l i u i -l i e g * u i x i ≤ ln(1 -α) + g * h, x ∈ X (31)
with arbitrary constant g * > 0.

Proof. Applying Bernstein inequality to the chance constraint in [START_REF] Andrieu | A model for dynamic chance constraints in hydro power reservoir management[END_REF], and passing to the logarithm both sides, the proof follows.

We provide now an upper bound to Problem (1) based on the Bernstein inequality.

Proposition 8. Given the random vectors ξ k for k = 1, . . . , K, having a joint distribution driven by the Gumbel-Hougaard copula C θ with some θ ≥ 1,

with components (ξ k ) i such that (l k ) i ≤ (ξ k ) i ≤ (u k ) i , and E[(ξ k ) i ] = ( ξk ) i ,
for k = 1, . . . , K and i = 1, . . . , n, an inner approximation of Problem [START_REF] Andrieu | A model for dynamic chance constraints in hydro power reservoir management[END_REF] can be obtained by solving the following problem min

x,y c T x s.t. n i=1 ln (u k ) i -( ξk ) i (u k ) i -(l k ) i e g * k (l k ) i x i + ( ξk ) i -(l k ) i (u k ) i -(l k ) i e g * i (u k ) i x i ≤ g * k h k + ln(1 -α y 1/θ k ), k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X, (32)
with arbitrary constants g * k > 0, k = 1, . . . , K.

Proof. According to Bernstein-type exponential estimate, the condition

e -g * k h k n i=1 (u k ) i -( ξk ) i (u k ) i -(l k ) i e g * k (l k ) i x i + ( ξk ) i -(l k ) i (u k ) i -(l k ) i e g * k (u k ) i x i ≤ α y 1/θ k , (33)
with arbitrary g

* k > 0, implies P( n i=0 (ξ k ) i x i ≥ h k ) ≤ α y 1θ k , k = 1, . . . , K. We note that P ξ T k x ≤ h k ≥ α y 1/θ k , k = 1, . . . , K, (34)
⇕ P n i=1 (ξ k ) i x i ≥ h k ≤ 1 -α y 1/θ k , k = 1, . . . , K. (35)
Problem [START_REF] Prékopa | Sharp bounds on probabilities using linear programming[END_REF] can be approximated as

n i=1 ln (u k ) i -( ξk ) i (u k ) i -(l k ) i e g * k (l k ) i x i + ( ξk ) i -(l k ) i (u k ) i -(l k ) i e g * k (u k ) i x i ≤ ln(1 -α y 1/θ k ) + g * k h k , for any g * k > 0, k = 1, . . . , K.
From Proposition 4.1 in [START_REF] Pinter | Deterministic approximations of probability inequalities[END_REF] and the concavity of function ln(1 -α y 1/θ k ), problem (32) is convex.

Hoeffding bounds

We consider now an approximation based on Hoeffding inequality ( [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]) given as follows:

P( ξ T e n n - ξT e n n ≥ h) ≤ e -2n 2 h 2 n i=1 (u i -l i ) 2 , (36)
with l i , u i the range parameters of the independent components ξ i of the random vector ξ, i.e.

l i ≤ ξ i ≤ u i , i = 1, . . . , n, ξ = E(ξ) and e n ∈ R n is a
vector with all elements equal to 1.

Proposition 9. With the assumption of ξ mentioned above, an inner approximation of Problem ( 2) can be obtained by solving the following convex problem min

x c T x s.t. ξT x + √ 2 2 -ln(1 -α)∥M x∥ ≤ h, x ∈ X, (37)
where

M = diag(u -l), u = (u 1 , . . . , u n ) T , l = (l 1 , . . . , l n ) T .
Proof. We note that

P ξ T x ≤ h ≥ α, (38)
⇕ P ξ T x -ξT x ≥ h -ξT x ≤ 1 -α. (39)
Then, we apply [START_REF] Prékopa | Probabilistic programming[END_REF] to [START_REF] Ruszczyński | Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra[END_REF] and get:

P ξ T x -ξT x ≥ h -ξT x ≤ e -2(h-ξT x) 2 n i=1 (u i -l i ) 2 x 2 i . ( 40
) If e -2(h-ξT x) 2 n i=1 (u i -l i ) 2 x 2 i ≤ 1 -α, (41)
then [START_REF] Prekopa | Programming under probabilistic constraint with discrete random variable[END_REF] will be satisfied. Logarithmic transformation of (41) leads to

-2(h -ξT x) 2 n i=1 (u i -l i ) 2 x 2 i ≤ ln(1 -α), (42)
which can be written as

h -ξT x ≥ √ 2 2 -ln(1 -α)∥M x∥, (43)
where M = diag(u -l) and then ( 43) is a convex inequality.

Proposition 10. Assume that the random vectors ξ k , k = 1, . . . , K have a joint distribution driven by the Gumbel-Hougaard copula C θ with some θ ≥ 1 and that the mean and the range parameters for all independent components (ξ

k ) i of ξ k are known, i.e. (l k ) i ≤ (ξ k ) i ≤ (u k ) i , for k = 1, . . . , K and i = 1, . . . , n.
An inner approximation of problem (1) based on Hoeffding inequality can be given by min

x,y c T x s.t. ξT k x + √ 2 2 ln 1 1 -α y 1/θ k ∥M k x∥ ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, x ∈ X, (44)
where

M k = diag(u k -l k ), u k = ((u k ) 1 , . . . , (u k ) n ) T , l k = ((l k ) 1 , . . . , (l k ) n ) T , k = 1, . . . , K.
Proof. With almost the same proof as Proposition 9, the conclusion can be obtained.

Additionally, with Assumption 4, an equivalent inner approximation of problem (1) based on Hoeffding inequality can be obtained by applying the following transformation z = ln x:

min z,y c T e z s.t. ξT k e z + 1 2 ∥M k e ln    2 ln   1 1-α y 1/θ k     +z ∥ ≤ h k , k = 1, . . . , K, K k=1 y k = 1, y k ≥ 0, k = 1, . . . , K, z ∈ Z. (45)
From Lemma 8, function ln

2 ln( 1 1-α y 1/θ k ) is convex, when α ≥ 1 - e -1 . Hence, if c ≥ 0, α ≥ 1 -e -1 , Problem (45) is convex.
When Assumption 4 and the convexity condition for problem [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF] do not hold, we should apply a sequential convex approximation algorithm similar to what proposed before for the Chebyshev bound.

A Refinery Blend Planning Problem under Uncertainty

In this section, we describe a problem of refinery operations which was originally proposed in [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF]. An important blend planning problem for a refinery consists of determining the optimal quantities x ip ∈ R + of material i ∈ B to blend together to obtain final products p ∈ P to sell on the market.

The refinery operation incurs a cost v i ∈ R + for producing or acquiring one unit of blendstock i ∈ B. On the other hand, selling the end product on the market yields a revenue f p ∈ R + , p ∈ P per unit. The objective of the refinery is to maximize the profit defined as the difference between revenues made from selling the end product and the cost of acquiring all blendstocks taking into account of the maximum production capacity mp of each product p ∈ P, a maximum flow rate mf i , a density d i of each blendstock i ∈ B, and a maximum target density dp of each product p ∈ P.

Typically, quality targets are imposed by law to prevent companies from selling substandard products which may damage engines and/or pollute the environment. We denote the target for quality k ∈ K in product p by t kp and let ζ ki the random value of quality k in blendstock i ∈ B. We assume that quality k ∈ K of a product p ∈ P is a linear function of the fraction of each blendstock used to produce it. Notice that the true qualities of blendstocks are actually unknown in real life applications. This means that if the operator considers ζ ki fixed to a nominal value ζki , the blending plan will lead to a significant loss since it will often yield off specification products which cannot be sold in the market. This issue implies that the refinery operator should explicitly account for uncertainties. In the following we consider a model which achieves this goal.

A probabilistic constrained programming formulation

In this section, we present a probabilistic constrained programming formulation of the refinery blend planning problem under uncertainty described above. We assume that the uncertain qualities ζ ki are modeled as random variables defined in the probability space (R K×n , F, P) which consists of the sample space R K×n , σ-algebra F and probability measure P.

ζ ki : value of quality k ∈ K in blendstock i ∈ B;
Decision variables:

x ip : quantity of blendstock i ∈ B to blend to obtain product p ∈ P.

The probabilistic constrained programming model for the blend planning problem is formulated as follows:

min p∈P i∈B (v i -f p )x ip (46)
s.t. P i∈B (ζ ki x ip ) ≤ t kp i∈B x ip , ∀k ∈ K ≥ α, ∀p ∈ P, (47)
i∈B

x ip ≤ mp, ∀p ∈ P, (48)
p∈P x ip ≤ mf i , ∀i ∈ B, (49)
i∈B d i x ip ≤ dp i∈B x ip , ∀p ∈ P, (50)
x ip ≥ 0, ∀i ∈ B, ∀p ∈ P.

The objective function [START_REF] Yoda | Improved bounds on the probability of the union of events some of whose intersections are empty[END_REF] expresses a minimization of the refinery costs defined as the difference between cost of acquiring all blendstocks and revenues made from selling. Constraints [START_REF] Zhang | Chance constrained programming for optimal power flow under uncertainty[END_REF] guarantee that each product p ∈ P is onspecification with probability greater than α only when all qualities k ∈ K jointly meet their targets. This generates the so-called joint probabilistic constraints. If these constraints cannot be satisfied, the resulting loss can be reflected in the objective function. In gasoline blending, for example, if octane rating is not well maintained in the product, it will damage the engine and may result in traffic problems. The fine, law sues, and safety issues cannot be easily quantified. Moreover, if unqualified product is produced in the refinery, the cost of compensation is significantly larger than the normal operations. How to estimate that cost and processing time is also challenging. The deterministic resource constraints ( 48)-( 49)-( 50) respectively impose a maximum production of each product, a maximum flow rate of each blendstock and a maximum target density of each product. Finally constraints (51) define the decision variables of the problem. Problem ( 46)-( 51) is typically noncovex and thus difficult to solve. A simple approach to deal with it, is to decompose the joint chance constraint [START_REF] Zhang | Chance constrained programming for optimal power flow under uncertainty[END_REF], for each product p ∈ P into K individual chance constraints as follows:

P i∈B (ζ ki x ip ) ≤ t kp i∈B x ip ≥ α, ∀p ∈ P, ∀k ∈ K, (52)
which corresponds to an inner approximation to constraint (47) according to Bonferroni inequality [START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF]. In the following we will solve problem ( 46)-(51) based on probabilistic inequalities derived in Section 3 which will enable us to obtain much less conservative approximations.

Numerical results

In this section, we provide an extensive numerical campaign to investigate the performance of the bounds proposed in Section 3 on the refinery blend planning problem under uncertainty ( 46)-( 51), with the aim of: understanding the performance of the considered inner approximations in terms of percentage GAP, CPU time and optimal blending recipes with respect to the exact SOCP reformulation for the single chance constraint case (see Subsection 5.1) and with respect to the Sample Average Approximations (SAA) method for the joint chance constraint case (see Subsection 5.2); reducing the computational complexity of the derived problems via a piecewise linear approximation based on tangent and segment approximation (see Appendix B); analyzing the sensitivity of the solution for increasing values of considered products qualities; analyzing the performance of the solutions obtained using the aforementioned bounds over the realization of different probability distributions (see Subsection 5.3).

The bounds have been implemented under Matlab R2018b environment using the CVX software, a modeling system for constructing and solving convex programs and SeDuMi solver. The computations have been performed on a 64-bit machine with 8 GB of RAM and a 1.8 GHz Intel i7 processor.

We first considered benchmark instances available in the literature (see [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF]) with some slight modifications as follows: by blending 10 types of intermediate flows B = {1, . . . , 10}, three types of gasoline P = {Type-1, Type-2, Type-3} are produced. The maximum production of each type of gasoline is mp = 50 and the set of qualities to be met is K = {RVP, (RON+MON)/2, Sulfur, Benzene}. We assume that the random variable ζ ki is distributed according to a normal distribution with mean ζki and standard deviation σ ki reported in tables 5-6 available in the Supplementary Material, respectively. Under this assumption, chance constraints can be equivalently reformulated as Second Order Cone Programming (SOCP) constraints as in (4). This will allow us to make a fair comparison of the bounds with the exact SOCP reformulation. Tables 7-8 in the Supplementary Material report the deterministic parameters mf i , v i , d i , with i ∈ B and dp , f p , with p ∈ P. Finally the targets t kp for quality k ∈ K in product p ∈ P are reported in Table 9. For both the individual and joint chance constraints cases, we choose a target probability α = 0.95. Typical values for an oil company are in the range α ∈ [95%; 99%] (see [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF]).

Secondly, we generate a set of instances extending the number of qualities up to |K| = 100. The new data (expected values ζki and standard deviations σ ki ) are uniformly generated respectively in the intervals [START_REF] Ahmed | Relaxations and approximations of chance constraints under finite distributions[END_REF]125] and (0, 1) and reported in Supplementary Material (see Tables 10,11 and 12). Although the problems which give upper bounds obtained by Chebyshev, Chernoff and Hoeffding inequalities for problem (1) are convex under some conditions, they are still hard to solve directly by current tools because of the following term: ln 2 ln( 1

1-α y 1/θ k ) .
In Appendix B we propose piecewise linear approximations for this function based on tangent and segment approximations as in [START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF].

The individual chance constrained case

We first consider model ( 46)-(51) for the blend planning problem where for any product p ∈ P the probabilistic constraints (47) are replaced by K individual chance constraints as in equation (52). Table 1 exhibits a comparison of the objective function optimal values and percentage gaps of the four bounds (Chebyshev, Chernoff, Hoeffding, and Bernstein) versus SOCP formulation assuming a normal distribution, as well as the CPU time (given in seconds) considering the data reported in Tables 56789. As upper and lower bounds (u ki and l ki ) for random vector ζ ki are needed in Bernstein and Hoeffding bounds, we generate 3000 samples following normal distributions with means ζki and standard deviations σ ki specified before. The maximal values of these 3000 samples are selected as upper bounds while the minimal values as lower bounds.

We first refer to the case of a collection of |K|×|P| = 12 individual chance constraints. Results show that the best upper bound is obtained by Chernoff reformulation with a percentage gap of only 0.74%, followed respectively by Chebyshev, Bernstein and Hoeffding having a percentage gap of 14.05%. In terms of CPU time, the most expensive is the Bernstein bound while the others are relatively comparable with SOCP. blending recipes x ip for the three types of gasolines (Type-1, Type-2, Type-3) over the ten blendstocks, are shown in Figures 1(a)-2 in Supplementary Material, for the SOCP formulation and all the considered bounds respectively.

SOCP Chebyshev

Results show that the most similar compositions to SOCP solution (0 for Type-1, 50 for Type-2 and 38.43 for Type-3) are given by Chernoff followed by Chebyshev ones, suggesting not to produce Type-1 and to saturate the production of Type-2, while they slightly underestimate the production of Type-3 with only 36.88 and 27.38 units respectively. Different is the composition obtained with Bernstein (0.20 for Type-1, 50 for Type-2 and 14.32 for Type-3) and Hoeffding solutions (5.55 for Type-1, 50 for Type-2 and 12.68 for Type-3) where Type-1 is suggested to be produced and the production of Type-3 is strongly underestimated.

We secondly refer to the case of a collection of |K| × |P| = 3K individual chance constraints with |K| = 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 qualities and |P| = 3 products. A comparison between the results of the exact SOCP formulation and its approximations (Chebyshev, Chernoff, Hoeffding, and Bernstein) is reported in Table 13 in Supplementary Material. Specifically, these results are obtained by considering the data corresponding to the first K rows of Tables 10, [START_REF] Dentcheva | On convex probabilistic programming with discrete distributions[END_REF] and 12 (see Supplementary Material).

As in the case of |K| = 4, numerical results confirm that the Chernoff approximation outperforms the others, as its percentage gaps vary in the interval 0.25%-10.30% increasing with an increasing number of qualities |K|. On the other hand, gaps for Chebyshev approximations range between 0.83%-34.98%, while gaps for Hoeffding approximations span between 1.53%-44.69%. A comparison of the percentage gap values with respect to SOCP formulation for increasing number of qualities |K| = 5, . . . , 100 is shown in Figure 4 in Supplementary Material. Results confirm the dominance of Chernoff bound, followed by Chebyshev and then by Hoeffding and the increasing behavior for considering a lager number of qualities |K|. The optimal total costs and corresponding percentage gaps of Bernstein bound have been obtained only in the case of one and five qualities with percentage gaps of respectively 2.51% (|K| = 1) and 30.50% (|K| = 5). For larger values (|K| ≥ 10) the approximated problem via Bernstein becomes infeasible via CVX software. Albeit Bernstein approximation seems performing better that the Hoeffding approximation, there is not enough empirical evidence to support this conclusion because of lack of numerical results for larger number of qualities.

Finally Table 13 and Figure 5 in Supplementary Material show the CPU time of Chebyshev,Chernoff and Hoeffding bounds for increasing number of qualities |K| = 1, . . . , 100. Bernstein approximation CPU time is omitted since considerably larger than the one required by the others (and as said before we cannot compute the bound for more than |K| = 5). CPU time for both Chebyshev and Chernoff approximations increases with the number of qualities in a similar way reaching a maximum of 7 CPU seconds, while Hoeffding approximation takes a maximum of only 1.81 CPU seconds with |K| = 100. Results suggest to use Hoeffding bound only when we need a rough solution in short time at expenses of low precision.

Joint chance constraint case

We investigate now the performance of the bounds for the blend planning problem with joint probabilistic constraints based on model ( 46)-(51).

Notice that, the blending problem cannot be solved according to the proposed logarithmic transformation for the following reasons: because of equation [START_REF] Zhang | Chance constrained programming for optimal power flow under uncertainty[END_REF], after applying the logarithmic transformation on x ip , the set Z for the new variable z is not convex. This is in contradiction with Assumption 1. Besides, the convexity condition for problems ( 14)-( 29)-( 45) does not hold, being c < 0. Therefore, the Chebyshev, Chernoff and Hoeffding bounds will be computed via a sequential convex approximation algorithm described above.

Besides, a Sample Average Approximation (SAA) method (see [START_REF] Pagnoncelli | Sample average approximation method for chance constrained programming: theory and applications[END_REF]) to solve the original joint chance constrained model ( 46)-(51), has been adopted. We now study the performance of the four bounds versus SAA assuming a normal distribution for ζ ki , k ∈ K, i ∈ B. As for the individual chance constraints case, as upper and lower bounds (u ki and l ki ) for random vector ζ ki are needed in Bernstein and Hoeffding bounds, we generate 3000 samples following normal distributions with means ζki and standard deviations σ ki specified before. The maximal values of these 3000 samples are selected as upper bounds while the minimal values as lower bounds. Besides, 1000 samples are generated according to the corresponding normal distribution of ζ ki for SAA method.

As before, we first refer to the case of joint chance constraints with |K| = 4 and |P| = 3 considering the data reported in Tables 56789. Results in Table 2 refer to the independent case (θ = 1). They show that the best upper bound is obtained by Chernoff reformulation with a percentage gap of only 0.24%, followed respectively by Bernstein, Chebyshev and Hoeffding having a percentage gap of 14.48%. The CPU time required by all the four bounds is considerably lower than the one required by SAA. This is due to the fact that the good quality of SAA solution requires a large size sample with an important number of binary variables and a high computational effort as consequence. The corresponding blending recipes x ip for the three types of gasolines (Type-1, Type-2, Type-3) over the ten blendstocks, are shown in Figures 1(b)-3 in Supplementary Material for the SAA method and all the considered bounds respectively. Results show that the most similar compositions to SOCP solution (0 for Type-1, 50 for Type-2 and 36.27 for Type-3) are given by Chernoff one, suggesting not to produce Type-1 and to saturate the production of Type-2, while it slightly underestimates the production of Type-3 with only 35.39. Different is the composition obtained with Bernstein (0.22 for Type-1, 50 for Type-2 and 12.02 for Type-3) where a small quantity of Type-1 is suggested to be produced and the production of Type-3 is underestimated. Finally both Chebyshev solution (0 for Type-1, 50 for Type-2 and 7.75 for Type-3) and Hoeffding solution (0 for Type-1, 50 for Type-2 and 4.13 for Type-3) match the correct quantities of Type-1 and Type-2 but strongly underestimate the one of Type-3.

SAA

A comparison of the bounds and SAA in the dependent case are shown in Table 3 for different values of θ = 2, . . . , 20. The quality of the bounds is similar as for the independent case, i.e., Chernoff followed by Chebyshev, Bernstein and Hoeffding. As done for the single chance constrained case, we extend now the investigation for the independent case, considering a larger number of qualities up to |K| = 100 with 3 products i.e., |P| = 3. Table 14 in Supplementary Material shows the comparison between the results of SAA and its approximations (Chebyshev, Chernoff, Hoeffding, and Bernstein) for, respectively, |K| = 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Specifically, the results are obtained by considering the data corresponding to the first K rows of Tables 10, [START_REF] Dentcheva | On convex probabilistic programming with discrete distributions[END_REF] and 12 (see Supplementary Material). Similarly to the individual case, the results in Table 14 show that the Chernoff approximation outperforms the others, as its percentage gaps vary in the interval 0.25%-11.02% and it increases with an increasing number of qualities. On the other hand, gaps for Chebyshev and Hoeffding bounds reach a percentage gap of 100% when |K| ≥ 20. The optimal total costs and corresponding percentage gaps for Bernstein bound can be obtained only in the case of five qualities with percentage gaps of 25.63%. This is because, when |K| ≥ 10, the approximated problem based on Bernstein bound becomes infeasible via CVX software. The gaps for Hoeffding approximations span between 42.98%-66.26% when |K| = 5, 10 while it becomes 100% when |K| ≥ 20.

θ SAA Chebyshev Chernoff
A comparison of the percentage gap values with respect to SAA for increasing number of qualities |K| = 5, . . . , 100 is shown in Figure 6 in Supplementary Material. Results confirm the dominance of Chernoff bound, followed by Bernstein for K = 5, and then by Hoeffding and Chebyshev. The increasing behavior for increasing numbers of qualities is also confirmed.

Results show that for large number of qualities, only Chernoff bound is appropriated in a joint chance constraint context, while the other approximations provide very loose bounds for large number of joint probabilistic conditions.

Finally Table 14 and Figure 7 in Supplementary Material show the CPU time of Chebyshev,Chernoff and Hoeffding bounds for increasing number of qualities |K| = 5, . . . , 100. Bernstein approximation CPU time is omitted since the approximated problem based on Bernstein bound is infeasible when K ≥ 10. CPU time for both Chebyshev and Hoeffding approximations increases with the number of qualities in a similar way reaching a maximum of 7.72 CPU seconds for 100 qualities, while Chernoff approximation takes a maximum of 13.11 CPU seconds with 10 qualities. The most remarkable result is that for |K| ≥ 5, the CPU time required by all the four bounds is considerably lower than the one required by SAA (see Table 14). In case of large number of joint probabilistic constraints, results suggest to use Chernoff bound while the other ones only when we need a rough solution in short time at expenses of low precision.

The error of the bounds

In this section we analyze the performance of the solutions obtained using the aforementioned bounds over the realization of different truncated probability distributions (see [START_REF] Maggioni | The value of the right distribution in stochastic programming with application to a newsvendor problem[END_REF]).

Let x ξ be the solution obtained by solving problem (2) using the probability distribution ξ and z ξ (x ξ ) be its optimal objective function value.

Let x bound be the solution obtained by solving problem (2) using one of the considered bounds (Chebyshev, Chernoff, Hoeffding and Bernstein) and let z ξ (x bound ) the corresponding objective function value. Notice that z ξ (x bound ) provides an upper bound of z ξ (x ξ ). We define the Percentage Error of the Bound %EB as

%EB := z ξ (x bound ) -z ξ (x ξ ) z ξ (x ξ ) • 100. ( 53
)
The greater is %EB, the greater is the objective function value increase of using the solution x bound when ξ occurs. When x bound is not feasible under distribution ξ we set %EB = ∞.

In Table 4 we compute %EB using one of the bounds (Chebyshev, Chernoff, Hoeffding and Bernstein via SAA) while the distribution ξ can be Cauchy, Logistic, Laplace and T-Student (with degree of freedom respectively equal to 2 and 3). Notice that the aforementioned distributions have been truncated in the same interval [l ki , u ki ] specified for Bernstein and Hoeffding bounds. Results refer to the case of a collection of |K| × |P| = 12 individual chance constraints. Chebyshev bound performs very well having %EB ∈ [3.07%, 3.32%]. Excellent is the performance of the Chernoff bound when ξ follows a Laplace or a T-Student (with degree of freedom equal to 3) with %EB ∈ [0.16%, 0.34%] while is infeasible when a Cauchy, a Logistic or T-Student (with degree of freedom equal to 2) occur. Larger are the errors using Bernstein bound solution having %EB ∈ [7.30%, 7

Conclusions

In this paper, we study deterministic inner approximations for single and joint probabilistic constraints. The derived upper bounds are based on classical inequalities from probability theory such as the one-sided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality. We show that under mild assumptions, the bounds based approximations of joint chance constrained problems are convex and the approximated problem can be solved efficiently. When the convexity condition can not hold, we show how to apply an efficient sequential convex approximation approach to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. To the best of our knowledge, these results are new in the literature since the majority of the contributions deals with symmetric elliptical distributions while we do not require any particular assumption on probability distributions.

Interesting results were also obtained by the computational experiments we carried out on a refinery blend planning problem under uncertainty. Comparing the bounds in terms of objective function, blending receipts and CPU time with respect to an exact SOCP formulation or SAA method, we found that the Chernoff approximation outperforms the others, both in case of individual and joint probabilistic constraints. In the individual case, results suggest to use Bernstein approximation only for a limited number of probabilistic constraints while to use Hoeffding bound when we need a rough solution in short time at expenses of low precision. A remarkable result in the joint case is that the CPU time required by all the four bounds is considerably lower than the one needed by SAA which requires to solve a difficult mixed integer program with binary variables. Finally a comparison of the error of using the solutions obtained by the bounds over different truncated probability distributions, has confirmed the good performance of Chebyshev bound followed by Bernstein and finally by Hoeffding. Chernoff bound should be considered more carefully having excellent error bounds with Laplace and T-Student (with degree of freedom equal to 3) while is infeasible for all the other considered probability distributions.

Future developments include the investigation of bounds for joint nonlinear chance constraints. This could be relevant to better model the nonlinear mixing laws of a blending problem, where for example, viscosity and octane blending are highly nonlinear.

Segment approximation

In order to come up with conservative bounds for the optimum values of problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and (45), we use the linear segments ās y k + bs , s = 1, . . . , S, between τ 1 , τ 2 , . . . , τ S+1 ∈ [ρ, 1) to construct a piecewise linear function l(y k ) = max s=1,...,S ās y k + bs ,

where ās = Υ(τ s+1 ) -Υ(τ s ) τ s+1 -τ s , bs = Υ(τ s ) -ās τ s , s = 1, . . . , S.

Using the piecewise linear function l(y k ) to replace Υ(y k ) in problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], gives the corresponding approximation problems.

Similar to Proposition 11, we can derive the following result for the linear approximation: Proposition 12. Under the aforementioned convex conditions, if we replace in problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and ( 45) Υ(y k ) by l(y k ), we obtain the convex approximations of these problems.

The optimum values of the approximation problems are an upper bound for problems [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], respectively. Moreover, the approximation problems become asymptotically an equivalent reformulation of problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], respectively, when S goes to infinity.

The proof of this Proposition follows the same pattern as the proof of Proposition 11.

Proposition 4 .

 4 If Assumptions 4 and 6 hold, then problem (14) is convex for all α ∈ [0, 1].

  Let us now define the following notation: Sets: B := {1, . . . , n}: set of materials to blend (blendstocks); P := {1, . . . , P }: set of final products to produce; K := {1, . . . , K}: set of qualities; Deterministic parameters: v i : cost for producing or acquiring one unit of blendstock i ∈ B; f p : revenue for selling one unit of product p ∈ P; t kp : target for quality k ∈ K in product p ∈ P; mp: maximum production of each type of gasoline; mf i : maximum flow of each blendstock i ∈ B; d i : density of each blendstock i ∈ B; dp : maximum target density for product p ∈ P; ζki : expected value of quality k ∈ K in blendstock i ∈ B; α: target probability; Uncertain parameters:

Figure 1 :

 1 Figure 1: Optimal blending recipes x ip for the three types of gasolines for (a) SOCP and (b) SAA formulations for the refinery blending planning problem with |K| × |P| = 12 individual and joint independent chance constraints respectively (|P| = 3 and |K| = 4).

  ζki

Figure 2 :

 2 Figure 2: Optimal blending recipes x ip for the three types of gasolines for SOCP formulation and all the considered bounds for the refinery blending planning problem with |K| × |P| = 12 individual chance constraints as in (52).

Figure 3 :

 3 Figure 3: Optimal blending recipes x ip for the three types of gasolines for all the considered bounds for the refinery blend planning problem with joint independent chance constraints (|P| = 3 and |K| = 4).

Figure 4 :

 4 Figure 4: % GAP of the bounds with respect to SOCP formulation for the refinery blending planning problem with 3K individual chance constraints with K = 5, . . . , 100.

Figure 5 :

 5 Figure 5: CPU time required by the different bounds for the refinery blending planning problem with 3K individual chance constraints with K = 5, . . . , 100.

Table 1 :

 1 Comparison of SOCP formulation versus bounds for the refinery blending planning problem with |K| × |P| = 12 individual chance constraints as in (52).

	Chernoff Hoeffding Bernstein

Table 2 :

 2 Comparison of SAA method versus bounds for the refinery blending planning problem with joint independent chance constraints with |K| = 4 and |P| = 3.

			Chebyshev Chernoff Hoeffding Bernstein
	Optimal value -767.56	-682.80	-765.69	-656.36	-714.57
	GAP (%)	-	11.04%	0.24%	14.48%	6.90%
	CPU time	152023	2.9	4.5	3.4	2

Table 3 :

 3 Comparison of SAA method versus bounds for the refinery blending planning problem with joint chance constraints with |K| = 4 and |P| = 3 for different θ.

Table 4 :

 4 .55%]. Worse is the performance of Hoeffding bound solution having %EB ∈ [15.83%, 16.05%]. Percentage Error of the Bound (%EB) obtained solving problem (2) using one of the bounds (Chebyshev, Chernoff, Hoeffding and Bernstein) while the distribution ξ (among truncated Cauchy, truncated Logistic, truncated Laplace and truncated T-Student with degree of freedom respectively equal to 2 and 3) occurs.

	Bounds

Table 10 :

 10 Expected value ζki of quality k ∈ K in blendstock i ∈ B.

	σ ki	1	2	3	4	5	6	7	8	9	10
	Quality 1	0.0333	0.6569	0.3487	0.5053	0.3533	0.2854	0.4984	0.5990	0.3368	0.1889
	Quality 2	0.9428	0.0930	0.9699	0.5173	0.1267	0.4141	0.6606	0.6161	0.3154	0.4452
	Quality 3	0.6615	0.3144	0.6641	0.6492	0.3878	0.2814	0.1650	0.0289	0.6019	0.3753
	Quality 4	0.4221	0.5639	0.8367	0.4091	0.9673	0.7182	0.4576	0.7663	0.9683	0.5440
	Quality 5	0.0460	0.7361	0.8123	0.9214	0.4598	0.0405	0.5800	0.4473	0.7305	0.8700
	Quality 6	0.2737	0.5691	0.2033	0.0824	0.9303	0.0114	0.0165	0.1831	0.8422	0.5907
	Quality 7	0.5929	0.7701	0.1963	0.4749	0.0902	0.9051	0.6335	0.2437	0.8264	0.5979
	Quality 8	0.9956	0.5674	0.3659	0.3257	0.4931	0.9409	0.2838	0.3709	0.9550	0.1287
	Quality 9	0.7363	0.4086	0.8641	0.7689	0.4371	0.5115	0.2985	0.4879	0.2445	0.6851
	Quality 10	0.0377	0.2399	0.6854	0.7733	0.2496	0.0227	0.8833	0.2893	0.6451	0.0660
	Quality 11	0.8024	0.4930	0.7012	0.7372	0.5751	0.8220	0.8201	0.2489	0.1335	0.7319
	Quality 12	0.6124	0.2711	0.6206	0.2371	0.2535	0.9411	0.9240	0.8942	0.4910	0.3811
	Quality 13	0.0325	0.9004	0.5909	0.8833	0.1435	0.4177	0.4116	0.5720	0.2792	0.0980
	Quality 14	0.7429	0.7671	0.9523	0.1472	0.9280	0.8428	0.8806	0.0203	0.0660	0.8309
	Quality 15	0.7861	0.3307	0.3714	0.9569	0.7326	0.2048	0.4278	0.8758	0.2420	0.7396
	Quality 16	0.0041	0.9788	0.7330	0.4686	0.2052	0.1111	0.4831	0.5717	0.2355	0.1559
	Quality 17	0.5462	0.8554	0.8626	0.4640	0.1178	0.9143	0.4319	0.2333	0.1506	0.1369
	Quality 18	0.4079	0.0298	0.5066	0.9365	0.6543	0.2693	0.2475	0.2605	0.0076	0.3115
	Quality 19	0.1087	0.9489	0.4444	0.5722	0.6732	0.1606	0.3227	0.8662	0.7111	0.5713
	Quality 20	0.8438	0.0500	0.3592	0.0362	0.9408	0.2178	0.5940	0.4764	0.5453	0.9064
	Quality 21	0.0340	0.3293	0.0041	0.2338	0.8450	0.2268	0.0464	0.6244	0.3712	0.2674
	Quality 22	0.0103	0.5189	0.2806	0.3052	0.3952	0.6677	0.8258	0.3017	0.4867	0.5713
	Quality 23	0.0329	0.6717	0.2023	0.2635	0.8311	0.8382	0.7809	0.3080	0.1408	0.9272
	Quality 24	0.5984	0.4712	0.2057	0.4445	0.9471	0.2909	0.1996	0.3804	0.6291	0.7125
	Quality 25	0.5461	0.0937	0.7449	0.3835	0.9020	0.0249	0.8419	0.6725	0.7168	0.3197
	Quality 26	0.3738	0.8215	0.9610	0.1671	0.3049	0.0625	0.9581	0.2106	0.7718	0.4092
	Quality 27	0.3019	0.2386	0.1475	0.3455	0.4950	0.5201	0.8820	0.4415	0.1587	0.2958
	Quality 28	0.0243	0.2430	0.4572	0.4646	0.1962	0.0273	0.3150	0.7659	0.8530	0.8030
	Quality 29	0.4201	0.4001	0.1417	0.2019	0.9770	0.9072	0.4916	0.7111	0.3220	0.7300
	Quality 30	0.0954	0.1283	0.5917	0.4021	0.1455	0.9161	0.6477	0.3540	0.4172	0.0184
	Quality 31	0.4211	0.9161	0.4367	0.3937	0.0828	0.7003	0.9229	0.6666	0.4174	0.6039
	Quality 32	0.2477	0.1341	0.6600	0.2643	0.5551	0.1339	0.9350	0.3984	0.4407	0.0842
	Quality 33	0.4021	0.2000	0.6598	0.2055	0.2038	0.3188	0.8659	0.1788	0.1854	0.0261
	Quality 34	0.4148	0.1389	0.9436	0.0096	0.7686	0.2046	0.7646	0.6890	0.6017	0.1839
	Quality 35	0.7036	0.6499	0.9894	0.0255	0.7185	0.4853	0.1215	0.7202	0.5509	0.4088
	Quality 36	0.5006	0.7891	0.3007	0.7905	0.4175	0.8432	0.1352	0.3075	0.5155	0.6112
	Quality 37	0.5010	0.4581	0.3033	0.9125	0.0884	0.6201	0.4941	0.5239	0.6900	0.7752
	Quality 38	0.2255	0.4625	0.1650	0.6039	0.1613	0.2811	0.9490	0.6819	0.6354	0.7922
	Quality 39	0.9156	0.8472	0.3825	0.1977	0.2701	0.4032	0.7736	0.7733	0.2920	0.1409
	Quality 40	0.4384	0.9931	0.0520	0.4561	0.8753	0.6049	0.0146	0.9235	0.7517	0.7000
	Quality 41	0.9179	0.5798	0.0125	0.4138	0.9256	0.7911	0.6629	0.6507	0.4778	0.4015
	Quality 42	0.9982	0.8910	0.8406	0.6447	0.1977	0.4663	0.9613	0.8388	0.2450	0.9860
	Quality 43	0.2504	0.3048	0.1599	0.4070	0.2567	0.9257	0.6421	0.1954	0.0662	0.9646
	Quality 44	0.1541	0.3685	0.7339	0.9454	0.6394	0.4436	0.3093	0.0407	0.5600	0.5232
	Quality 45	0.6458	0.7752	0.5346	0.6233	0.9799	0.6155	0.2815	0.3584	0.6043	0.1654
	Quality 46	0.6043	0.7481	0.4322	0.0764	0.1137	0.7924	0.3126	0.3524	0.2595	0.3995
	Quality 47	0.4753	0.9274	0.1562	0.5070	0.9312	0.4318	0.4909	0.6604	0.1478	0.5071
	Quality 48	0.1246	0.9934	0.6797	0.0048	0.4979	0.0758	0.3949	0.8198	0.7284	0.0603
	Quality 49	0.9630	0.4216	0.6314	0.2995	0.4942	0.6002	0.2119	0.5924	0.6292	0.4539
	Quality 50	0.9361	0.0581	0.7038	0.8725	0.0226	0.6633	0.0173	0.2646	0.8423	0.4105
	Quality 51	0.2441	0.6092	0.6722	0.9156	0.3682	0.9263	0.0776	0.2513	0.9081	0.5468
	Quality 52	0.7519	0.2599	0.7745	0.5755	0.5236	0.4304	0.4321	0.2519	0.0249	0.2250
	Quality 53	0.4887	0.9625	0.4735	0.6786	0.7118	0.8561	0.3076	0.3286	0.2978	0.8358
	Quality 54	0.0456	0.2964	0.4966	0.3688	0.3375	0.9977	0.2508	0.2223	0.1666	0.9787
	Quality 55	0.8683	0.6455	0.8483	0.2994	0.0324	0.1010	0.3874	0.8631	0.8256	0.2095
	Quality 56	0.1784	0.3359	0.1689	0.3926	0.3946	0.2085	0.0411	0.7044	0.3232	0.2578
	Quality 57	0.3452	0.5581	0.6045	0.4286	0.5995	0.1252	0.4089	0.0295	0.6652	0.7693
	Quality 58	0.8231	0.5737	0.2325	0.8416	0.6836	0.7604	0.7338	0.0982	0.8486	0.2388
	Quality 59	0.5595	0.3472	0.2648	0.9428	0.0714	0.4674	0.0545	0.8824	0.9779	0.9224
	Quality 60	0.4044	0.9961	0.3863	0.4102	0.9954	0.4215	0.7425	0.5270	0.4323	0.5217

Table 13 :

 13 Comparison of SOCP formulation versus bounds for the refinery blending planning problem with 3K individual chance constraints as in (52) with K = 1, 10, . . . , 100.

		SOCP	Chebyshev	Chernoff	Hoeffding	Bernstein
			K = 1
	Optimal value	-965.34	-957.36	-962.97	-950.57	-941.13
	GAP (%)	-	0.83%	0.25%	1.53%	2.51%
	CPU time	0.72	0.66	0.67	0.50	2.03
			K = 10
	Optimal value	-481.23	-387.48	-453.56	-266.16	-
	GAP (%)	-	19.48%	5.75%	44.69%	-
	CPU time	1.17	1.08	1.39	1.22	68.50
			K = 20
	Optimal value	-432.45	-340.23	-407.42	-245.08	-
	GAP (%)	-	21.32%	5.79%	43.33%	-
	CPU time	2.78	2.30	2.08	0.88	198.52
			K = 30
	Optimal value	-431.59	-302.97	-400.58	-250.29	-
	GAP (%)	-	29.80%	7.18%	42.01%	-
	CPU time	2.89	2.67	2.56	0.95	176.53
			K = 40
	Optimal value	-422.79	-298.04	-396.14	-259.23	-
	GAP (%)	-	29.51%	6.30%	38.69%	-
	CPU time	2.73	2.44	2.36	1.11	217.66
			K = 50
	Optimal value	-422.79	-298.04	-396.14	-257.58	-
	GAP (%)	-	29.51%	6.30%	39.08%	-
	CPU time	3.06	3.03	2.88	0.95	309.38
			K = 60
	Optimal value	-422.79	-297.80	-396.14	-250.94	-
	GAP (%)	-	29.56%	6.30%	40.65%	-
	CPU time	3.75	3.55	3.94	1.22	313.34
			K = 70
	Optimal value	-421.56	-280.52	-383.10	-254.58	-
	GAP (%)	-	33.46%	9.12%	39.61%	-
	CPU time	4.36	4.14	4.69	1.67	327.58
			K = 80
	Optimal value	-401.91	-261.34	-360.53	-237.24	-
	GAP (%)	-	34.98%	10.30%	40.97%	-
	CPU time	6.78	6.22	5.27	1.56	410.38
			K = 90
	Optimal value	-401.91	-261.34	-360.53	-247.43	-
	GAP (%)	-	34.98%	10.30%	38.44%	-
	CPU time	5.41	5.19	5.33	1.67	572.45
			K = 100
	Optimal value	-401.91	-261.34	-360.53	-249.29	-
	GAP (%)	-	34.98%	10.30%	37.97%	-
	CPU time	6.03	7.23	6.17	1.81	476.98

Acknowledgments

The authors wish to thank the Associate Editor and the review team whose comments led to an improved version of this paper.

Appendix A

Proof of Lemma 5 Proof. Since f is convex, we have that for λ ∈ [0, 1] and x 1 , x 2 ∈ X, f (λx 1 + (1 -λ)x 2 ) ≤ λf (x 1 ) + (1 -λ)f (x 2 ). Therefore, as g : Y → Z be a nonincreasing concave function, we have

which proves the thesis.

Proof of Lemma 7 Proof. Since e tξ T x is convex with respect to x ∈ X, we have that for λ ∈ [0, 1] and x 1 , x 2 ∈ X, e tξ T (λx 1 +(1-λ)x 2 ) ≤ λe tξ T x 1 + (1 -λ)e tξ T x 2 . Therefore, E(e tξ T (λx 1 +(1-λ)x 2 ) ) ≤ E(λe tξ T x 1 +(1-λ)e tξ T x 2 ) = λE(e tξ T x 1 )+(1-λ)E(e tξ T x 2 ), which proves the thesis.

Proof of Lemma 8 Proof. We only need to prove the convexity of the function b → ln 2 ln( ) . Therefore, we only need to focus on the convexity of b → ln 2 ln( 11-b ) . We have

As

Tangent approximation

We choose S different linear functions:

Here ρ ≥ 0 is a constant such that Υ(y k ) is convex on [ρ, 1). Then, Υ(y k ) can be approximated by the following piecewise linear function

which provides a lower approximation for Υ(y k ).

In order to achieve the expected precision, we set l s (y k ) as the tangent line of Υ(y k ) at S points τ 1 , . . . , τ S with τ s ∈ [ρ, 1), s = 1, . . . , S. Then, we have

Thanks to these piecewise linear approximations for Υ(y k ), we have the following results: Proposition 11. Under the aforementioned convex conditions, if we replace in problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and (45) Υ(y k ) by l(y k ), we obtain their convex approximations. The optimum values of the approximation problems are lower bounds for problems [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], respectively. Moreover, the approximation problems become asymptotically an equivalent reformulation of problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and (45) when S goes to infinity.

Proof. As the approximation problems are obtained by relaxing some constraints in problems ( 14), ( 29), [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], it is easy to see that the optimal values of the approximation problems are lower bounds for problems [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], ( 29), [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], respectively.

We know under convex conditions for problems [START_REF] Geletu | Advances and applications of chance-constrained approaches to systems optimisation under uncertainty[END_REF], [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and (45), Υ(y k ) is convex for each problem. As the S tangent functions are selected differently, when S goes to infinity, the constraints in the approximation problems are asymptotically equivalent to the constraints in problems ( 14), [START_REF] Ouarda | Chance-constrained optimal control for multireservoir system optimization and risk analysis[END_REF], and [START_REF] Yang | Chance-constrained optimization for refinery blend planning under uncertainty[END_REF], respectively.