Convexity of chance constraints with dependent random variables: the use of copulae

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1 Convexity of chance constraints with dependent random variables: the use of copulae René Henrion 1 and Cyrille Strugarek 2 1 Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany. henrion@wias-berlin.de 2 Credit Portfolio Management, Calyon Credit Agricole CIB, Paris, France cyrille.strugarek@calyon.com Summary. We consider the convexity of chance constraints with random righthand side. While this issue is well understood thanks to Prékopa s Theorem if the mapping operating on the decision vector is componentwise concave, things become more delicate when relaxing the concavity property. In an earlier paper, the significantly weaker r-concavity concept could be exploited, in order to derive eventual convexity starting from a certain probability level for feasible sets defined by chance constraints. This result heavily relied on the assumption of the random vector having independent components. A generalization to arbitrary multivariate distributions is all but straightforward. The aim of this paper is to derive the same convexity result for distributions modelled via copulae. In this way, correlated components are admitted, but a certain correlation structure is imposed through the choice of the copula. We identify a class of copulae admitting eventually convex chance constraints. Acknowledgement. The work of the first author was supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin 1 Introduction Chance constraints or probabilistic constraints represent an important tool for modeling restrictions on decision making in the presence of uncertainty. They take the typical form Phx, ξ 0 p, 1 where x R n is a decision vector, ξ : Ω R m is an m-dimensional random vector defined on some probability space Ω, A, P, h : R n R m R s is a vector-valued mapping and p [0, 1] is some probability level. Accordingly, a decision vector x which is subject to optimization with respect to some given objective function is declared to be feasible, if the random inequality system hx, ξ 0 is satisfied at least with probability p. A compilation of practical applications in which constraints of the type 1 play a crucial role, may be found in the standard references [Pre95],

2 2 René Henrion and Cyrille Strugarek [Pre03]. Not surprisingly, one of the most important theoretical questions related to such constraints is that of convexity of the set of decisions x satisfying 1. In this paper, we shall be interested in chance constraints with separated random vectors, which come as a special case of 1 by putting hx, ξ = gx ξ. More precisely, we want to study convexity of a set of feasible decisions defined by Mp = {x R n Pξ gx p}, 2 where g : R n R m is some vector-valued mapping. With F : R m R denoting the distribution function of ξ, the same set can be rewritten as Mp = {x R n Fgx p}. 3 It is well-known from Prékopa s classical work see, e.g., [Pre95], Th that this set is convex for all levels p provided that the law P ξ 1 of ξ is a log-concave probability measure on R m and that the components g i of g are concave. The latter condition is satisfied, of course, in the important case of g being a linear mapping. However, the concavity assumption may be too strong in many engineering applications, so the question arises, whether it can be relaxed to some weaker type of quasi- concavity. To give an example, the function e x is not concave, while its log is. It has been shown in an earlier paper that such relaxation is possible indeed, but it comes at the price of a rather strong assumption of ξ having independent components and of a slightly weaker result, in which convexity of Mp cannot be guaranteed for any level p but only for sufficiently large levels. We shall refer to this property as eventual convexity. Note, however, that this is not a serious restriction in chance constrained programming, because probability levels are chosen there close to 1 anyway. To provide an example see Example 4.1 in [HS08], let the two dimensional random vector ξ have a bivariate standard normal distribution with independent components and assume that g ix 1, x 2 = 1/x x i = 1, 2. Then, for p 0.7 one has that Mp is nonconvex for p < p whereas it is convex for p p. Note, that though the multivariate normal distribution is log-concave see [Pre95], the result by Prékopa mentioned above does not apply because the g i are not concave. As a consequence, Mp is not convex for all p as it would be guaranteed by that result. Nonetheless, eventually convexity can be verified by the tools developed in [HS08]. The aim of this paper is to go beyond the restrictive independence assumption made in [HS08]. While to do this directly for arbitrary multivariate distributions of ξ seems to be very difficult, we shall see that positive results can be obtained in case that this distribution is modeled by means of a copula. Copulae allow to represent dependencies in multivariate distributions in a more efficient way than correlation does. They may provide a good approximation to the true underlying distribution just on the basis of its one-dimensional margins. This offers a new perspective also to modeling chance constraints not considered extensively so far to the best of our knowledge. The paper is organized as follows: in a first section, some basics on copulae are presented and the concepts of r-concave and r-decreasing functions introduced. The following section contains our main result on eventual convexity of chance constraints defined by copulae. In a further section, log-exp concavity of copulae, a decisive property in the mentioned convexity result, is discussed. Finally,

3 Convexity of chance constraints: the use of copulae 3 a small numerical example illustrates the application of the convexity result in the context of chance constrained programming. 2 Preliminaries 2.1 Basics on copulae In this section, we compile some basic facts about copulae. We refer to the introduction [Nel06] as a standard reference. Definition 1. A copula is a distribution function C : [0, 1] m [0, 1] of some random vector whose marginals are uniformly distributed on [0, 1]. A theorem by Sklar states that for any distribution function F : R m [0, 1] with marginals F i 1 i m there exists a copula C for F satisfying x R m, Fx = C F 1x 1,... F mx m. 4 Moreover, if the marginals F i are continuous, then the copula C is uniquely given by Cu = F F 1 1 u 1,..., Fm 1 u m ; F 1 i t := inf r. 5 F i r t Note that Sklar s Theorem may be used in two directions: either in order to identify the copula of a given distribution function or to define a distribution function via a given copula for instance a copula with desirable properties could be used to fit a given distribution function. The following are first examples for copulae: Independent or Product Copula: Cu = Π m u i Maximum or Comonotone Copula: Cu = min 1 i m u i Gaussian or Normal Copula: C Σ u = Φ Σ Φ 1 u 1,..., Φ 1 u m Here, Φ Σ refers to a multivariate standard normal distribution function with mean zero, unit variances and covariance =correlation matrix Σ. Φ is just the onedimensional standard normal distribution function. With regard to 4, the independent copula simply creates a joint distribution with independent components from given marginals. The theoretical interest of the maximum copula also explaining its name comes from the fact that any copula C is dominated by the maximum copula: Cu min ui. 6 1 i m The Gaussian copula has much practical importance similar to the normal distribution function. It allows to approximate a multivariate distribution function F having marginals F i which are not necessarily normally distributed, by a composition of a multivariate normal distribution function with a mapping having components Φ 1 F i defined via one-dimensional distribution functions [KW97]. A lot of other practically important copulae is collected in a whole family:

4 4 René Henrion and Cyrille Strugarek Definition 2. A copula C is called Archimedean if there exists a continuous strictly decreasing function ψ : [0, 1] R + such that ψ1 = 0 and m Cu = ψ 1 ψu i, ψ is called the generator of the Archimedean copula C. If lim u 0 ψu = +, then C is called a strict Archimedean copula. Conversely, we can start from a generator ψ to obtain a copula. The following proposition provides sufficient conditions to get a copula from a generator: Proposition 1. Let ψ : [0, 1] R + such that 1. ψ is strictly decreasing, ψ1 = 0, lim u 0 ψu = + 2. ψ is convex 3. 1 k dk dt k ψ 1 t 0 k = 0, 1,..., m, t R +. Then, Cu = ψ 1 m ψui is a copula. Example 1. We have the following special instances of Archimedean copulae: 1. The independent copula is a strict Archimedean copula whose generator is ψt = logt. 2. Clayton copulas are the Archimedean copulas defined by the strict generator ψt = θ 1 t θ 1, with θ > Gumbel copulas are the Archimedean copulas defined by the strict generator ψt = logt θ, with θ Frank copulas are the Archimedean copulas defined by the strict generator ψt = log e θt 1 e θ 1, with θ > r-concavity and r-decreasing density functions We recall the definition of an r-concave function see, e.g., [Pre95]: Definition 3. A function f : R s 0, is called r-concave for some r [, ], if fλx + 1 λy [λf r x + 1 λf r y] 1/r x, y R s, λ [0, 1]. 7 In this definition, the cases r {, 0, } are to be interpreted by continuity. In particular, 1-concavity amounts to classical concavity, 0-concavity equals logconcavity i.e., concavity of log f, and -concavity identifies quasi-concavity this means that the right-hand side of the inequality in the definition becomes min{fx, fy}. We recall, that an equivalent way to express log-concavity is the inequality fλx + 1 λy f λ xf 1 λ y x, y R s, λ [0, 1]. 8 For r < 0, one may raise 7 to the negative power r and recognize, upon reversing the inequality sign, that this reduces to convexity of f r. If f is r -concave, then f is r-concave for all r r. In particular, concavity implies log-concavity. We shall be mainly interested in the case r 1. The following property will be crucial in the context of this paper:

5 Convexity of chance constraints: the use of copulae 5 Definition 4. We call a function f : R R r-decreasing for some r R, if it is continuous on 0, and if there exists some t > 0 such that the function t r ft is strictly decreasing for all t > t. Evidently, 0-decreasing means strictly decreasing in the classical sense. If f is a nonnegative function like the density of some random variable, then r-decreasing implies r -decreasing whenever r r. Therefore, one gets narrower families of r- decreasing density functions with r. If f is not just continuous on 0, but happens even to be differentiable there, then the property of being r-decreasing amounts to the condition tf t + rft < 0 for all t > t. It turns out that most one-dimensional distribution functions of practical use share the property of being r-decreasing for any r > 0 an exception being the the density of the Cauchy distribution which is r-decreasing only for r < 2. The following table, borrowed from [HS08] for the readers convenience, compiles a collection of one-dimensional densities all of which are r-decreasing for any r > 0 along with the respective t -values from Definition 4 labeled as t r to emphasize their dependence on r: Table 1. t r- values in the definition of r-decreasing densities for a set of common distributions. Law Density t r normal t µ2 µ+ 1 2πσ exp 2σ 2 exponential λ exp λt t > 0 µ 2 +4rσ 2 2 r λ b+r 1 1/b ab Weibull abt b 1 exp at b t > 0 b Gamma a exp bt Γa ta 1 a+r 1 t > 0 b 1 χ 2 n/2 1 Γn/2 tn 1 exp t2 t > 0 n + r 1 2 χ n/2 Γn/2 tn/2 1 exp t t > 0 n + 2r log t µ2 log-normal 2πσt exp t > 0 e µ+r 1σ2 2σ 2 2t Maxwell 2 2πσ 3 exp t2 t > 0 σ r + 2 2σ 2 2t Rayleigh exp t2 r+1 t > 0 λ λ λ 2 We shall need as an auxiliary result the following lemma whose proof is easy and can be found in [HS08] Lemma 3.1: Lemma 1. Let F : R [0, 1] be a distribution function with r + 1-decreasing density f for some r > 0. Then, the function z Fz 1/r is concave on 0, t r, where t refers to Definition 4. Moreover, Ft < 1 for all t R. 3 Main Result The purpose of our analysis is to investigate convexity of the set

6 6 René Henrion and Cyrille Strugarek Mp = {x R n C F 1g 1x,..., F mg mx p }. 9 Here, g is as in 3, the F i are the one-dimensional marginals of some m-dimensional distribution function F and C is a copula. If C is the exact copula associated with F via 4 or 5, respectively, then the feasible set 9 coincides with the chance constraint 3. In general, we rather have in mind the idea that F is unknown and C serves its approximation. Then, 9 can be understood as a chance constraint similar to 3 or 2 but with the distribution function F occuring there replaced by the distribution function C H, where H ix := F ix i for i = 1,..., m. We introduce the following property of a copula which will be crucial for our convexity result: Definition 5. Let q 0, 1 m. A copula C : [0, 1] m [0, 1] is called log exp-concave on Π m [q i, 1 if the mapping C : R m R defined by is concave on Π m [log q i, 0. Cu = log C e u 1,..., e um, 10 The following statement of our main result makes reference to Defintions 3, 4, 5. Theorem 1. In 9, we make the following assumptions for i = 1,..., m: 1. There exist r i > 0 such that the components g i are r i-concave. 2. The marginal distribution functions F i admit r i + 1-decreasing densities f i. 3. The copula C is log exp-concave on Π m [F it i, 1, where the t i refer to Definition 4 in the context of f i being r i + 1-decreasing. Then, Mp in 9 is convex for all p > p := max{f it i 1 i m}. Proof. Let p > p, λ [0, 1] and x, y Mp be arbitrary. We have to show that λx + 1 λy Mp. We put q x i := F ig ix < 1, q y i := Figiy < 1 i = 1,..., m, 11 where the strict inequalities rely on the second statement of Lemma 1. In particular, by 11, the inclusions x, y Mp mean that Cq x 1,..., q x m p, Cq y 1,..., q y m p. 12 Now, 11, 6, 12 and the definition of p entail that 1 > q x i p > F it i 0, 1 > q y i p > Fit i 0 i = 1,..., m. 13 For τ [0, 1], we denote the τ-quantile of F i by F iτ := inf{z R F iz τ}. Note that, for τ 0, 1, Fiτ is a real number. Having a density, by assumption 2., the F i are continuous distribution functions. As a consequence, the quantile functions F iτ satisfy the implication q > F iz = F iq > z q 0, 1 z R. Now, 11 and 13 provide the relations

7 Convexity of chance constraints: the use of copulae 7 g ix F iq x i > t i > 0, g iy F iq y i > t i > 0 i = 1,..., m 14 note that t i > 0 by Definition 4. In particular, for all i = 1,..., m, it holds that [ min{ F r i i qi x, F r i i q y i }, max{ F r i i qi x, F ] r i i q y i } 0, t i r i. 15 Along with assumption 1., 14 yields for i = 1,..., m: g i λx + 1 λy λg r i i λ F r i i x + 1 λg r i i y 1/r i q x i + 1 λ F r i i q y i 1/ri. 16 The monotonicity of distribution functions allows to continue by F i g i λx + 1 λy F i λ F r i i qi x + 1 λ F r i i q y i 1/ri i = 1,..., m. 17 Owing to assumption 2., Lemma 1 guarantees that the functions z F iz 1/r i are concave on 0, t i r i. In particular, these functions are log-concave on the indicated interval, as this is a weaker property than concavity see Section 2.2. By virtue of 15 and 8, this allows to continue 17 as [ ] λ [ ] 1 λ F i g i λx + 1 λy F i Fiq i x F i Fiq y i i = 1,..., m. Exploiting the fact that the F i as continuous distribution functions satisfy the relation F i F iq = q for all q 0, 1, and recalling that qi x, q y i 0, 1 by 13, we may deduce that F i g i λx + 1 λy [q x i ] λ [q y i ]1 λ i = 1,..., m. Since the copula C as a distribution function is increasing w.r.t. the partial order in R m, we obtain that C F 1g 1λx + 1 λy,..., F mg mλx + 1 λy C [q1] x λ [q1] y 1 λ,..., [qm] x λ [qm] y 1 λ 18 According to assumption 3. and 13, we have log C [q1] x λ [q1] y 1 λ,..., [qm] x λ [qm] y 1 λ = log C explog[q1] x λ [q1] y 1 λ,..., explog[qm] x λ [qm] y 1 λ = log C expλlog [q x 1] + 1 λ log [q y 1],..., expλlog [q x m] + 1 λ log [q y m] λlog Cq x 1,..., q x m + 1 λ log Cq y 1,..., q y m λlog p + 1 λ log p = log p, where the last inequality follows from 12. Combining this with 18 provides C F 1g 1λx + 1 λy,..., F mg mλx + 1 λy p. Referring to 9, this shows that λx + 1 λy Mp.

8 8 René Henrion and Cyrille Strugarek Remark 1. The critical probability level p beyond which convexity can be guaranteed in Theorem 1, is completely independent of the mapping g, it just depends on the distribution functions F i. In other words, for given distribution functions F i, the convexity of Mp in 2 for p > p can be guaranteed for a whole class of mappings g satisfying the first assumption of Theorem 1. Therefore, it should come at no surprise that, for specific mappings g even smaller critical values p may apply. 4 Log exp-concavity of copulae Having a look to the assumptions of Theorem 1, the first one is easily checked from a given explicit formula for the mapping g, while the second one can be verified, for instance, via Table 1. Thus, the third assumption, namely log exp-concavity of the copula, remains the key for applying the Theorem. The next proposition provides some examples for log exp-concave copulae: Proposition 2. The independent, the maximum and the Gumbel copulae are log exp-concave on [q,1 m for any q > 0. Proof. For the independent copula we have that m m log C e u 1,..., e um = log e u i = and for the maximum copula it holds that log C e u 1,..., e um = log min 1 i m eu i = min 1 i m log eu i = min 1 i m ui. Both functions are concave on, 0 m, hence the assertion follows. For the Gumbel copula see Example 1, the strict generator ψ t := log t θ for some θ 1 implies that ψ 1 s = exp s 1/θ, whence, for u i, 0, m m log C e u 1,..., e um = log ψ 1 ψ e u i = log ψ 1 u i θ u i m 1/θ m 1/θ = log exp u i θ = u i θ = u θ. Since θ is a norm for θ 1 and since a norm is convex, it follows that θ is concave on, 0 m. Thus, the Gumbel copula too is log exp-concave on [q, 1 m for any q > 0. Contrary to the previous positive examples, we have the following negative case: Example 2. The Clayton copula is not log exp-concave on any domain m [qi, 1 where q 0, 1 m. Indeed, taking the generator ψ t := θ 1 t θ 1, we see that ψ 1 s = θs + 1 1/θ and calculate

9 Convexity of chance constraints: the use of copulae 9!! m m X X log C eu1,..., eum = log ψ 1 ψ eui = log ψ 1 θ 1 e θui 1 = θ 1 log 1 m+ m X e θui!. Now, for any t < 0, define ϕ t := log C et,..., et = θ 1 log 1 m + me θt. Its second derivative calculates as θm m 1 eθt. m m 1 eθt 2 Q m Now, if C was log exp-concave on any m [qi, 1 where q 0, 1, then, in particular, ϕ would be concave on the interval [τ, 0, where τ := max log qi < 0. This ϕ t =,...,m however, contradicts the fact that ϕ t > 0 for any t. Fig. 1. Plot of the function log Cex, ey for the bivariate Gaussian copula C in case of positive left and negative right correlation Concerning the Gaussian copula it seems to be difficult to check log exp-concavity even in the bivariate case. At least, numerical evidence shows that the bivariate Gaussian copula is log exp-concave for non-negative correlation coefficient but fails to be so for negative correlation coefficient. This fact is illustrated in Figure 1, where the thick curve in the right diagram represents a convex rather than concave piece of the function log Cex, ey. The next proposition provides a necessary condition for a copula to be log expconcave. It relates those copulae with the well-known family of log-concave distribution functions: Q Proposition 3. If a copula C is log exp-concave on m [qi, 1, then it is log concave on the same domain.. Proof. By definition of log exp-concavity, we have that

10 10 René Henrion and Cyrille Strugarek log C exp λy λ z 1,..., exp λy m + 1 λ z m λlog C exp y 1,..., exp y m + 1 λ log C exp z 1,..., exp z m 19 for all λ [0, 1] and all y, z m [log qi, 0. Now, let λ [0, 1] and u, v m [qi, 1 be arbitrary. By concavity of log and monotonicity of exp, we have that exp log λu i + 1 λ v i exp λlog u i + 1 λ log v i i = 1,..., m. 20 Since C as a distribution function is nondecreasing with respect to the partial order of R m, the same holds true for log C by monotonicity of log. Consequently, first 20 and then 19 yield that log C λu + 1 λ v = log C exp log λu λ v 1,..., exp log λu m + 1 λ v m log C exp λlog u λ log v 1,..., exp λlog u m + 1 λ log v m Hence, C is log concave on m [qi, 1. λlog C u 1,..., u m + 1 λ log C v 1,..., v m = λlog C u + 1 λ log C v. 5 An example A small numerical example shall illustrate the application of Theorem 1. Consider a chance constraint of type 2: P ξ 1 x 3/4 1, ξ 2 x 1/4 2 p. Assume that ξ 1 has an exponential distribution with parameter λ = 1 and ξ 2 has a Maxwell distribution with parameter σ = 1 see Table 1. Assume that the joint distribution of ξ 1, ξ 2 is not known and is approximated by means of a Gumbel copula C with parameter θ = 1. Then, the feasible set defined by the chance constraint above is replaced by the set 9 with F 1, F 2 being the cumulative distribution functions of the exponential and Maxwell distribution, respectively, and with g 1 x 1, x 2 = x 3/4 1, g 2 x 1, x 2 = x 1/4 2. When checking the assumptions of Theorem 1, we observe first, that the third one is satisfied due to Proposition 2. Concerning the first assumption, note that the components g 1 and g 2 fail to be concave they are actually convex. According to Section 2.2, the g i would be at least r-concave, if for some r < 0 the function gi r was convex. In our example, we may choose r = 4/3 for g 1 and r = 4 for g 2. Thus, we have checked the first assumption of the Theorem with r 1 = 4/3 and r 2 = 4. It remains to verify the second assumption. This amounts to require the density of the exponential distribution to be 7/3-decreasing and the density of the Maxwell distribution to be 5-decreasing. According to Table 1 and with the given parameters of these distributions, these properties hold true in the sense of Definition 4 with a t -value of t 1 = r 1 + 1/λ = 7/3 in case of the exponential distribution and with a t -value of t 2 = σ r = 7 in case of the Maxwell distribution. Now, Theorem 1 guarantees convexity of the feasible set 9, whenever p > p, where

11 Convexity of chance constraints: the use of copulae Fig. 2. Illustration of the feasible set Mp in 9 for different probability levels in a numerical example. Convexity appears for levels of approximately 0.8 and larger } p = max {F 1 7/3, F 2 7. Using easily accessible evaluation functions for the cumulative distribution functions of the exponential and Maxwell distribution, we calculate F 1 7/ and F Hence, Mp in 9 is definitely convex for probability levels larger than Figure 2 illustrates the resulting feasible set for different probability levels feasible points lie below the corresponding level lines. By visual analysis, convexity holds true for levels p larger than approximately 0.8. Not surprisingly, our theoretical result is more conservative. One reason for the gap is explained in Remark 1. References [HS08] Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl., 41, [KW97] Klaassen, C.A.J., Wellner, J.A.: Efficient Estimation in the Bivariate Normal Copula Model: Normal Margins are Least Favourable, Bernoulli, 3, [Nel06] Nelsen, R. B.: An Introduction to Copulas. Springer, New York 2006 [Pre95] Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht 1995 [Pre03] Prékopa, A.: Probabilistic Programming. In: Ruszczynski, A. and Shapiro, A. eds.: Stochastic Programming. Hamdbooks in Operations Research and Management Science, Vol. 10. Elsevier, Amsterdam 2003