On the Convexity of Level-sets of Probability Functions
Yassine Laguel, Wim van Ackooij, Jérôme Malick, Guilherme Ramalho
aa r X i v : . [ m a t h . O C ] F e b On the Convexity of Level-sets of Probability Functions
Yassine Laguel Wim Van Ackooij Jérôme Malick Guilherme Matiussi Ramalho Univ. Grenoble Alpes, CNRS, LJK, 38000 Grenoble, France EDF R&D, Saclay, France Federal University of Santa Catarina, LABPLAN, Santa Catarina, Brazil
Abstract
In decision-making problems under uncertainty, probabilistic constraints are a valuable toolto express safety of decisions. They result from taking the probability measure of a given setof random inequalities depending on the decision vector. Even if the original set of inequali-ties is convex, this favourable property is not immediately transferred to the probabilisticallyconstrained feasible set and may in particular depend on the chosen safety level. In this paper,we provide results guaranteeing the convexity of feasible sets to probabilistic constraints whenthe safety level is greater than a computable threshold. Our results extend all the existingones and also cover the case where decision vectors belong to Banach spaces. The key idea inour approach is to reveal the level of underlying convexity in the nominal problem data (e.g.,concavity of the probability function) by auxiliary transforming functions. We provide severalexamples illustrating our theoretical developments.
We consider a probabilistic constraint built up from the following ingredients: a map g : X × R m → R k , where X is a (reflexive) Banach space, a random vector ξ ∈ R m defined on an appropriateprobability space, and a user-defined safety level p ∈ [0 , . The probabilistic constraint then reads: ϕ ( x ) := P [ g ( x, ξ ) ≤ ≥ p , (1)where ϕ : X → [0 , is the associated probability function. The interpretation of (1) is simple: onerequires the decision x to be such that the random inequality system g ( x, ξ ) ≤ holds with prob-ability at least p . Such constraints (also called chance-constraint) often appear in decision-makingproblems under uncertainty; for the theory and applications of chance-constraints optimization, werefer to [8, 20, 22, 49, 63] and references therein.In this paper we focus on the “convexity of the probabilistic constraint" (1), i.e., the convexityof the set of feasible solutions defined by M ( p ) := { x ∈ X : P [ g ( x, ξ ) ≤ ≥ p } . (2)Understanding when M ( p ) is a convex set is important for the point of view of optimization, toguarantee that local solutions are also globally optimal and to use numerical solution methods thatexploit this convexity (we review the most popular methods in section 1.3). A first result of the1onvexity of M ( p ) follows from Prékopa’s celebrated log-concavity theorem (see [18, Proposition 4]for its infinite dimensional version and [12] for generalizations): the convexity of M ( p ) is guaran-teed for all p ∈ [0 , , when − g is jointly quasi-concave in both arguments and ξ an appropriaterandom vector. However, joint-quasi-concavity of − g is rather “exceptional" and fails in many ba-sic situations. For example, when g ( x, ξ ) = x T ξ and ξ is multi-variate Gaussian, it is well knownthat M ( p ) is convex only whenever p ≥ (see e.g., [28]). In this example and many others, wethus observe that if the convexity of M ( p ) does not hold for all p , there still exists a (computable)threshold p ∗ ∈ [0 , such that the set M ( p ) is convex for all p ≥ p ∗ . This property is called eventualconvexity as observed by [48] and coined by [24] (which studies the case where g is separable and ξ has independent components). Eventual convexity results are further generalized in [25] by allowingfor the components of ξ to be coupled through a copulæ dependency structure. These results arerefined, by allowing for more copulæ and with sharper bounds for p ∗ in [56], and extended to allArchimedian copulæ in [59], where also an appropriate solution algorithm is provided. When themapping g is non-separable, eventual convexity results are provided in [65] for the special case where ξ is elliptically symmetrically distributed. Here we will simplify, clarify and extend these results. In this paper, we build on this line of research about establishing convexity of superlevel-sets ofprobability functions (2) for p larger than a threshold. We show that a notion of generalizedconcavity naturally appears in this framework, and allows us to reveal the level of hidden convexityof the data. We formalize a way to analyze separately the convexity inherent to the randomnessand the one associated with the optimization model structure.Roughly speaking, our approach is the following. In various contexts, the probability functioninvolves a composition of two functions F ◦ Q , with F : R → [0 , carrying the randomness of theproblem and Q : R n → R given by the optimization model. We split the problem of establishingconcavity (or at least quasi-concavity) of the composition F ◦ Q by finding an adequate function G (the inverse of which is denoted G − ) to write F ◦ Q = F ◦ G − ◦ G ◦ Q (3)such that F ◦ G − and G ◦ Q satisfy appropriate convexity properties. Thus this approach naturallyraises interest in the concavity of function G ◦ Q , which is a “transformable concavity", formalizedby the notion called transconcavity or G -concavity by [53] and [2]. Similarly, we will briefly studyconcavity of terms F ◦ G − and introduce the counterpart notion that we will call concavity- G − .Beyond this intuition, the analysis of the convexity of chance-constrained sets is not trivial: F ,as a distribution function, cannot be concave and one has to be careful with working on appropriatesmaller subsets. The size of those subsets turns out to be directly related to the level of probabilitybeyond which convexity is guaranteed. Arguments of this type are implicitly and partially used in[24, 56, 59, 65]. Here we highlight this approach, exploit it in its full generality, and provide a setof tools to apply it in practice. We apply this set of tools to get eventual convexity statements intwo general contexts(A) when ξ is specific (elliptically distributed) and g is general (with light geometrical assump-tions),(B) when ξ is general (with known associated copula) and g is specific (of the form g ( x, ξ ) = ξ − h ( x ) ) 2n these two cases, we prove the existence of the threshold p ∗ such that M ( p ) is convex for all p ≥ p ∗ .Beyond the theoretical contribution regarding the existence of p ∗ , we also provide a concrete wayof numerically evaluating p ∗ from the nominal data. Our results are illustrated through variousexamples (not captured by existing results) of eventual convexity with a specified threshold p ∗ .The contributions of this work are thus the following ones. Our main contribution is to identifythe interplay of generalized concavity with the functions used to model the constraints and theuncertainty in chance-constrained optimization problems. This clarification would allow practition-ers to refine their models of nonlinearity and readily swapping certain uncertainty distributions byothers sharing similar generalized concavity properties. Since we care about practical use of ourresults, this introduction is meant to be an accessible overview of the state-of-the-art on eventualconvexity and its use in practice. We also provide througout the paper many examples illustratingour results. Finally we specifically list our main technical contributions:• We slightly extend the existing notion of G -concavity (by allowing for decreasing functions G )and introduce the new notion of concavity- G − that goes with it. We also extend a useful resultof [24] on one-dimensional distribution functions and give necessary and sufficient conditionsunder which such functions can be composed with monotonous maps to make them concave.• For the context (A), our results clarify and extend those of [65] allowing us to treat newsituations: we use various forms of G -concavity and thus can cover a wider range of non-linearmappings g .• For the context (B), we first refine the results of [68] by providing a better threshold. More im-portantly, our results cover new situations as they tackle the most general case with nonlinearmappings and copulæ, while [56, 59] restricts to α -concavity (a special G ) and [68] restrictsto independent copulæ.• We also extend all these previous results (that consider finite dimensional decision vectors x ),by analysing and stating our results in Banach spaces. This opens the door to cover recentapplications in PDE-constrained optimization (e.g., [18]).The paper is organized as follows. After section 1.3, which will put this work into a broaderpractical perspective, our development starts in Section 2, which provides a careful account of usefuland used notions of generalized concavity together with calculus rules. Section 3 then providesgeneral eventual convexity statements in the two contexts (A) and (B) in Sections 3.1 and 3.2respectively. Finally Section 4 is devoted to providing examples covered in our extended framework. Although this work is theoretical, it can be inserted in the bigger picture of solving chance-constrainedoptimization problems. Of course, a priori, evaluating probability functions is computationally de-manding (see e.g., the discussion in [45]), especially if the random vector is highly dimensional.Still recent results (e.g., [6]) indicate that random vectors with dimensions in their hundreds, i.e.,practically relevant sizes, can be handled with CPU times hovering roughly around a minute. ThoseCPU times relate to solving a non-convex optimization problem involving a probability function,evaluated repeatedly. Although this does not alleviate the theoretical or algorithmic difficulties,3t does show that, by exploiting the structure present in applications, one can solve applicationsof relevant size. The existing numerical solution methods use sample-based approximations of theprobabilistic constraint or treat probability functions (or a surrogate) by non-linear optimizationtechniques; we briefly review the main methods here. Notice that many of these methods rely onsome convexity properties of the chance-constrained problems, bringing interest to the results ofthis paper.Popular numerical methods for dealing with probabilistic constraints are sample-based approx-imations, e.g., [39, 40, 41, 46] with various strenghtening procedures, e.g., [32, 67] or investigationof convexification procedures [1]. We can also mention boolean approaches, e.g., [30, 31, 36, 37], p -efficient point based concepts, e.g., [9, 10, 11, 38, 58], robust optimization [4], penalty approach[14], scenario approximation [7, 50], convex approximation [45], or yet other approximations [19, 26].Aside from this rich literature, the non-linear constraint (1) can also be dealt with directly as such,from the study of (generalized) differentiability of probability functions and the development ofreadily implementable formulæ for gradients. Such formulæ can be further improved by using wellknown “variance reduction” techniques, such as Quasi-Monte Carlo methods (e.g., [5]) or importancesampling (e.g., [3]). For further insights on differentiability, we refer to e.g., [23, 29, 42, 51, 54, 55, 62].Nonlinear programming methods using these properties include sequential quadratic programming[6] and the promising bundle methods [59, 66].Practical probabilistic constrained problems also involve several other constraints, that can berepresented as an abstract subset S ⊂ X . Important questions concern, in fact, the constrained set M ( p ) ∩ S , for which the results presented in this paper might be used. As a brief observation wedo write p and not p ∗ , since p is the user chosen safety level and thus what is practically relevant.• Is M ( p ) ∩ S convex? Convexity of S is achieved in many practical cases: in a significant shareof applications S is polyhedral, or easily seen to be convex. The difficulty in establishingconvexity of M ( p ) ∩ S therefore lies in checking whether M ( p ) is a convex set, which is theaim of this paper. As a sufficient condition, the user would check p ≥ p ∗ , when applicable, inorder to have a global guarantee on computed solutions. Since, occasionally, p ∗ may dependadversely on random vector dimension, if the test fails, this does not necessarily imply that M ( p ) ∩ S is not convex. The results of this paper rather indicate that a computed “solution”should not necessarily be taken as a global solution, because convexity of M ( p ) ∩ S is nolonger guaranteed. Such an information is still useful for the user, who may decide to investadditional effort in running the local optimization solver with multiple starting points, orcalling another (expensive) global optimization solver.• Is M ( p ) ∩ S non-empty ? The safety level p is chosen by the user, who, as a modeler, isresponsible for ensuring that a well-posed model is formulated. In practice, we can expect areasonable convex optimization solver to return an infeasibility flag when the set M ( p ) ∩ S is empty (in the case that M ( p ) ∩ S is convex as attested by the previous point). In such acase, the user can examine his data, and subsequently formulate a better model. Answeringthe feasibility question without convexity is of course an entirely different matter, and sucha theoretically and algorithmically difficult problem goes largely beyond the scope and thesetting of this paper. Below we mention some relevant heuristic procedures that have workedwell in our experience.The feasibility regarding probabilistic restrictions is related to the question of maximizing the proba-bility function over X or S , which recently has received special attention; see e.g., [15, 16, 43]. Indeed,4f the maximal probability thus found is greater than or equal to p , then M ( p ) ∩ S is ensured to bea nonempty set. Of course, finding the global solution of this probability maximization problem is ahard task in general, because the probability function need not be concave. Heuristically, feasibilitycan also be addressed by considering a sample based variant of probabilistic maximization problem,with few samples, i.e., min z ,...,z N N X i =1 z i s.t. g ( x, ξ i ) ≤ M z i , x ∈ S, z i ∈ { , } , where M is an appropriate “big-M” constant, S is the deterministic constraints set, and ξ , ..., ξ N are i.i.d. samples of ξ . The last problem can in principle be solved with fairly few samples (small N ) and with low accuracy (e.g., 10% MIP-gap) to produce ¯ x . An a posteriori evaluation of theprobability function can then assert feasibility of ¯ x for the true probabilistic constraint. Indeed,when g is convex in x and S is convex, one can solve the last program with the methodology laidout in [61]. The sample can also be exploited in a “scenario approach” (the asymptotics with respectto N are well studied in for instance [50]). Finally, the approach (maximization of copula structuredprobability not requiring convexity) in [60] can also be employed. It consists of minimizing a lower- C function (requiring easily verified differentiability, whenever the copula is Archimedian) withtools from nonsmooth optimization. In this section, we gather the tools on generalized concavity that we will use in next sections to revealthe underlying concavity of nominal data in probabilistic constraints. Section 2.1 briefly reviews thedefinitions and useful properties of G -concavity (also called transconcavity (see [2])) and Section2.2 introduces the right counterpart of concavity- G − . We provide new technical lemmas, includinga characterization of generalized concavity of cumulative distribution functions. G -oncavity We start by recalling the notion of G -concavity introduced by [53] and presented in the book [2]under the name transconcavity. We just add here the possibility of G being strictly decreasing,which will turn out to be useful in our context. Definition 1.
Let X be a Banach space and C be a convex subset of X . We say that a function f : C → R is G -concave if there exists a continuous and strictly monotonic function G : f ( C ) → R such that f ( λx + (1 − λ ) y ) ≥ G − ( λG ◦ f ( x ) + (1 − λ ) G ◦ f ( y )) holds for all x, y ∈ C and λ ∈ [0 , . In this paper, we denote the inverse of a map G by G − , and the division by a map, whenever well defined, by G . G is increasing, G -concavity of f is just the concavity of the map G ◦ f . When G is decreasing, G -concavity of f is simply the convexity of G ◦ f . A given function f can be “ G -concave”for several different mappings G . It will be convenient, however, to pin down a specific choice andsubsequently speak of G -concavity of f for such a specific choice. The naming “transconcavity"would then refer to an unspecified, yet implicitly assumed to exist, mapping G for which f is G -concave. Example . A particularly well studied set of choices for G is that of the family G α : t t α for α ∈ R \ { } and G : t ln( t ) . (4)This family has several properties that help to measure a “level of generalized concavity” of a function f , as used below in the definition of α -concavity.We introduce the following mapping m α : R + × R + × [0 , → R (for a given α ∈ [ −∞ , ∞ ) )defined as follows: if ab = 0 and α ≤ , m α ( a, b, λ ) = 0 (5)else, for λ ∈ [0 , , we let: m α ( a, b, λ ) = a λ b − λ if α = 0min { a, b } if α = −∞ ( λa α + (1 − λ ) b α ) α else (6)This enables us to define the known notion of α -concavity as a “particular case” of G -concavity. Definition 2 ( α -concave function) . Let X be a Banach space and C be a convex subset of X . Wesay that a function f : C → R + is α -concave if f ( λx + (1 − λ ) y ) ≥ m α ( f ( x ) , f ( y ) , λ ) , (7)for all x, y ∈ C and λ ∈ [0 , .Observe that for α = 0 , α -concavity of f is indeed equivalent with G α -concavity of f (in thesense of Definition 1 with G α of (4)). Notice that -concavity coincides with the usual notion ofconcavity.We also note that our definition of the mapping m α differs slightly of that found in [8, Def 4.7],in so much that we have appended the condition α ≤ to condition (5). The reason for this is thatotherwise the definition does not match up with what is expected whenever a = 0 or b = 0 and α > . In particular consider α = 1 and the usual definition of concavity for a function f , two points x, y ∈ C with f ( y ) = 0 for instance. Since our definition slightly differs from the classical one, weprovide the proof of the following technical lemma used to establish the hierarchy of α -concavity. Lemma 3.
Let a, b ∈ R + , λ ∈ [0 , be given and fixed. Then for α, β ∈ [ −∞ , ∞ ) , m β ( a, b, λ ) ≤ m α ( a, b, λ ) holds when β ≤ α . Moreover the map α m α ( a, b, λ ) is continuous.Proof. Let a, b, λ be as in the statement. Since λ is arbitrary and m α ( a, b, λ ) = m α ( b, a, − λ ) , wemay without loss of generality assume a ≤ b . Furthermore, let also β ≤ α be given but fixed. Wewill proceed by a case distinction. 6 irst case : If α ≥ β > or α > > β then the mapping t t αβ is convex on R + . So, we have λu αβ + (1 − λ ) v αβ ≥ ( λu + (1 − λ ) v ) αβ , for u, v ≥ which since α > , and hence t t α strictly increasing on R + , is equivalent to ( λu αβ +(1 − λ ) v αβ ) α ≥ ( λu + (1 − λ ) v ) β . The desired result follows by substituting u = a β and v = b β . Second case : If > α ≥ β , then we have − β ≥ − α > and for any u, v ≥ , we can apply theprevious case to obtain the inequality: ( λu − β + (1 − λ ) v − β ) − β ≥ ( λu − α + (1 − λ ) v − α ) − α . The latter is equivalent with ( λ ( 1 u ) β + (1 − λ )( 1 v ) β ) β ≤ ( λ ( 1 u ) α + (1 − λ )( 1 v ) α ) α , provided that u, v > hold. Assuming a > (and hence b > ), we may substitute u = a and v = b to obtain the desired inequality. When ab = 0 , since both α and β ≤ , the desired inequalityholds. Third case : To treat a case where α = 0 or β = 0 , we first establish continuity of α m α around0. To this end, consider the following Taylor expansions: a α = e α ln( a ) = 1 + α ln( a ) + o ( α )1 α ln( λa α + (1 − λ ) b α ) = 1 α ln(1 + α ( λ ln( a ) + (1 − λ ) ln( b )) + o ( α )) = λ ln( a ) + (1 − λ ) ln( b ) + o (1) Consequently, we get at the limit when α → α ln( λa α + (1 − λ ) b α )) = exp( λ ln( a ) + (1 − λ ) ln( b ) + o (1))) → a λ b − λ = m ( a, b, λ ) , Now assume β = 0 and α > . If m ( a, b, λ ) ≤ m α ( a, b, λ ) were not to hold, then it would followthat m α ( a, b, λ ) < m ( a, b, λ ) . We may pick a sequence α k ↓ and for k large enough it holds < α k ≤ α . By the already established order, we have m α k ( a, b, λ ) ≤ m α ( a, b, λ ) < m ( a, b, λ ) .But this contradicts the just established continuity of α m α ( a, b, λ ) near . The situation α = 0 can be established along similar lines of argument. Fourth case : The situation wherein β < < α follows by invoking the third case twice, since indeed m β ( a, b, λ ) ≤ m ( a, b, λ ) ≤ m α ( a, b, λ ) . Last case : When β = −∞ . The inequality m −∞ ( a, b, λ ) ≤ m α ( a, b, λ ) holds trivially whenever ab = 0 . By combining the previous cases we may assume α < as well as a, b > . We observe that a, b ≥ m −∞ ( a, b, λ ) = min { a, b } and since α < , it holds a α , b α ≤ m −∞ ( a, b, λ ) α . Consequentlytoo, λa α + (1 − λ ) b α ≤ m −∞ ( a, b, λ ) α . Since t t α is strictly decreasing, we get m −∞ ( a, b, λ ) ≤ m α ( a, b, λ ) .We finish by proving the continuity. Since both terms λa α and (1 − λ ) b α are nonnegative, wehave that: max( λa α , (1 − λ ) b α ) ≤ λa α + (1 − λ ) b α , m −∞ ( a, b, λ ) ≤ ( λa α + (1 − λ ) b α ) α ≤ max( λa α , (1 − λ ) b α ) α = [max( λ − α a , (1 − λ ) − α b )] − . (8)Hence by passing to the limit: lim α →−∞ [max( λ − α a , (1 − λ ) − α b )] − = [max( 1 a , b )] − = min( a, b ) , This gives the continuity at −∞ .The above property of the map m α allows us to establish an entire hierarchy of “concavity”immediately, as formalized by the next corollary. Corollary 4 (Hierarchy of α -concavity) . Let X be a Banach space and C be a convex subset of X . Let the map f : C → R + , together with α, β ∈ [ −∞ , ∞ ) , be given. If f is α -concave, it is also β -concave when α ≥ β . In particular f is quasi-concave. The family of mappings { G α } α of (4) allows us to distinguish the level of generalized concavityof a function f ranging from quasi-concavity ( α = −∞ ) to classic concavity ( α = 1 ). Intuitively, thegreater α is, the “more concave”, f will be. All the practical examples in this paper will use thesefunctions to quantify and extract underlying convexity. Let us mention though that this familydoes not capture completely the subtle notion of transconcavity (see Example 2 below) and thatalternative families of functions could be considered, such as the exponential family of functions G r : t
7→ − e − rt , for varying values of r , extensively studied in [2, Chap.8]. Example α -concavity) . Let us provide an example of a mapping h : R → R + that is not α -concave for any α ∈ R , but is G -concave for an appropriate choice of amap G . We will show that h : R → R + defined as h ( x ) = exp( − x ) is not α -concave for any α < (and then by Corollary 4 that h can not be α -concave for any α ). Indeed, let α < be arbitrary.Then α -concavity of h is equivalent to convexity of h α . Now by differentiating twice we obtain: ddx ( h α )( x ) = − x α ( h α )( x ) and d dx ( h α )( x ) = h α ( x ) αx (9 αx − . For x < , αx > and moreover x αx − has unique (negative root) x r = ( α ) . Consequently αx − > for x < x r and αx − < for x > x r . By combining with the above, we establishthat d dx ( h α )( x ) < must hold for x ∈ ( x r , , implying that h α can not be convex, i.e., h is not α -concave.Now define G : (0 , + ∞ ) → R + as G ( x ) = exp( − (ln( x )) ) , then by direct computation ddx G ( x ) = − x (ln( x )) G ( x ) at any x = 1 . We now readily verify that G is strictly decreasing. Moreover, G ( h ( x )) = exp( x ) ; hence, by definition, h is G -concave.We now recall and extend a lemma from [2] that enables us to propagate the property of G -concavity. Lemma 5 (Propagation of generalized concavity) . Let X be a Banach space and C be a convexsubset of X . Let the map f : X → R be a G -concave function for an appropriate choice G : f ( C ) → R . Let G be a continuous and strictly monotonic function over f ( C ) . If G − , the inversefunction of G is G -concave over G ◦ f ( C ) , then f is also G -concave over C . roof. By assumption we have, for any x, y ∈ C, λ ∈ [0 , that f ( λx + (1 − λ ) y ) ≥ G − ( λG ◦ f ( x ) + (1 − λ ) G ◦ f ( y )) and for any u, v ∈ G ◦ f ( C ) , λ ∈ [0 , , we have G − ( λu + (1 − λ ) v ) ≥ G − ( λG ◦ G − ( u ) + (1 − λ ) G ◦ G − ( v )) . Hence, if we fix x, y ∈ C, λ ∈ [0 , and we set u = G ◦ f ( x ) , v = G ◦ f ( y ) , we get from thesetwo inequalities: f ( λx + (1 − λ ) y ) ≥ G − ( λu + (1 − λ ) v ) ≥ G − ( λG ◦ G − ( u ) + (1 − λ ) G ◦ G − ( v ))= G − ( λG ◦ G − ( G ◦ f ( x )) + (1 − λ ) G ◦ G − ( G ◦ f ( y ))) ≥ G − ( λG ◦ f ( x ) + (1 − λ ) G ◦ f ( y )) , which gives the result. G − We introduce in this section the notion of concavity- G − which is the right counterpart of theclassical G -concavity recalled previously. In view of (3), the two complementary notions will beuseful in the sequel, in particular because we establish that many cumulative distribution functionsare concave- G − . Along the way, we generalize a result of [24]. Definition 6 (concave- G − functions) . Let F : R → R and G : R → R be continuous and strictlymonotonic mappings. The map F is said to be concave- G − on an interval I ⊂ R if F ◦ G − isconcave on the interval I . By extension of Definition 6, we also speak of a G -concave- G − function F if G ◦ F ◦ G − is concave.This definition can be specialized as follows when considering the family { G α } α of Example 1.Let α ∈ ( −∞ , be given; we say that f : R → R is concave- α on an interval I ⊂ R if (i) f isincreasing and t f ( t α ) is concave on I , or (ii) f is decreasing and t f ( t α ) is convex on I .(Note that t f (exp( t )) has to be understood whenever α = 0 is chosen.) Let us provide a positiveexample of concave- α functions in our context. Example . Let F : R + → [0 , be the cumulative distributionfunction of a χ -random variable with m degrees of freedom. Then for any α , F is concave- α on theinterval I = (0 , ( m − α ) α ] if α < , I = [ln( m ) / , ∞ ) if α = 0 and I = [( m − α ) α , ∞ ) if α ∈ (0 , .This can be established by direct computation or as a result of [65, Lemma 3.1]. Further positiveexamples can be found in [24, Table 1].As we will shortly see, concavity- G − and concavity- α of cumulative distribution functions canbe conveniently related to specific properties of their density functions (provided they exist). Tothis end, we introduce the following concept. Definition 7 ( G -decreasing functions) . Let G : R + → R + be a strictly decreasing (resp. increasing)continuously differentiable map with finitely many critical points. A mapping f is said to be G -decreasing (resp. G-increasing) if there exists t ∗ G > [max (cid:8) t : G ′ ( t ) = 0 (cid:9) ] + such that the ratio r ( t ) := f ( t ) G ′ ( t ) is strictly decreasing (increasing) on the set t ≥ t ∗ G . Here [ t ] + :=max { t, } is the positive part. 9he instantiation of this definition, related to the family of mappings { G α } α was already intro-duced in [24] under the notion of α -decreasing functions (only considering the situation α < ). Wenow provide a key result relating concavity- G − of a given distribution function with its densitybeing G -decreasing. Proposition 8 (concave- G − cumulative distribution functions and G -decreasing densities) . Let F : R → [0 , be the cumulative distribution function of a random variable with associated (continuouslydifferentiable) density function f . Consider the statements:1. the density f is G -decreasing (see Definition 7) with associated parameter t ∗ ;2. the mapping F is concave- G − on the interval I = (0 , G ( t ∗ )] if G is strictly decreasing, andon I = [ G ( t ∗ ) , ∞ ) if G is strictly increasing, i.e., z F ( G − ( z )) is concave on I ;Then 1. implies 2. and if moreover G is twice continuously differentiable, then 2. also implies 1..Proof. ⇒ . We note that the proof of this implication follows closely the proof of [24, Lemma3.1] as well as [68, Lemma 4]. Let G : R + → R + be a strictly increasing (decreasing) map and t ∗ be such t f ( t ) G ′ ( t ) is strictly decreasing (increasing) on the set t ≥ t ∗ .Let us begin by considering the situation wherein G is strictly decreasing. Then for z ∈ (0 , G ( t ∗ )) ,we have G − ( z ) ≥ t ∗ . Now, the map z F ( G − ( z )) that we will call χ can be written: χ ( z ) = Z G − ( z ) −∞ f ( s ) ds = F ( t ∗ ) + Z G − ( z ) t ∗ f ( s ) ds = F ( t ∗ ) + Z zG ( t ∗ ) f ( G − ( u )) G ′ ( G − ( u )) du = F ( t ∗ ) − Z G ( t ∗ ) z f ( G − ( u )) G ′ ( G − ( u )) du, where we have carried out the substitution u = G ( s ) . The ratio appearing in the integral is acontinuous map, making χ (continuously) differentiable. Moreover, χ ′ ( z ) = f ( G − ( z )) G ′ ( G − ( z )) , which together with z ∈ (0 , G ( t ∗ )) implies G − ( z ) ≥ t ∗ so that χ ′ is strictly decreasing. As aconsequence χ is indeed concave.Let us now consider the case wherein G is strictly increasing. Then for z ∈ ( G ( t ∗ ) , ∞ ) it alsoholds that G − ( z ) ≥ t ∗ . We can write χ as χ ( z ) = Z G − ( z ) −∞ f ( s ) ds = F ( t ∗ ) + Z G − ( z ) t ∗ f ( s ) ds = F ( t ∗ ) + Z zG ( t ∗ ) f ( G − ( u )) G ′ ( G − ( u )) du where we have carried out the substitution u = G ( s ) . Now χ ′ ( z ) = f ( G − ( z )) G ′ ( G − ( z )) , which together with z ∈ ( G ( t ∗ ) , ∞ ) implies G − ( z ) ≥ t ∗ so that χ ′ is strictly decreasing. As a result, χ is concave. 10 . ⇒ Let us assume to begin with that F is concave- G − , on the interval [ G ( t ∗ ) , ∞ ) for a strictlyincreasing map G and define χ ( z ) = F ◦ G − ( z ) . We first note that G is strictly increasing and(continuously differentiable) and hence by the classic inverse function Theorem (e.g., [13, Theorem1A.1], G − is also continuously differentiable and the identity ( G − ) ′ ( x ) = G ′ ◦ G − ( x ) holds.Now, by assumption, χ is concave on [ G ( t ∗ ) , ∞ ) , and for any x ∈ [ G ( t ∗ ) , ∞ ) , we have χ ′ ( x ) = ( G − ) ′ ( x ) . ( f ◦ G )( x ) χ ′′ ( x ) = ( G − ) ′′ ( x ) . ( f ◦ G )( x ) + ( G − ) ′ ( x ) . ( f ′ ◦ G )( x ) . We can rewrite the second derivative as follows: χ ′′ ( x ) = 1( G ′ ◦ G − )( x ) (cid:18) − G ′′ ◦ G − ( x )( G ′ ◦ G − ( x )) f ◦ G − ( x ) + f ′ ◦ G − G ′ ◦ G − ( x ) (cid:19) , where we have used the identity ( G − ) ′′ ( x ) = − G ′′ ◦ G − ( x )( G ′ ◦ G − ( x )) resulting from differentiating twice in theidentity G ( G − ( x )) = x holding locally at any x > and thus in particular at any x ∈ [ G ( t ∗ ) , ∞ ) since G ( t ∗ ) > . Since G is strictly increasing, so is G − and consequently G ′ ◦ G − ( x ) > .Concavity of χ implies in turn that χ ′′ ( x ) ≤ , which can be equivalently stated as: − G ′′ ( t )( G ′ ( t )) f ( t ) + f ′ ( t ) G ′ ( t ) ≤ , (9)where t = G − ( x ) and x ≥ G ( t ∗ ) if and only if t ≥ t ∗ . By defining ψ : t f ( t ) G ′ ( t ) and differentiatingonce, we obtain: ψ ′ ( t ) = f ′ ( t ) G ′ ( t ) + f ( t )( − G ′′ ( t ) G ′ ( t ) = − G ′′ ( t ) G ′ ( t ) f ( t ) + f ′ ( t ) G ′ ( t ) . Now by (9) it follows that ψ ′ ( t ) ≤ for all t ≥ t ∗ and hence by Definition 7, f is G -decreasing.This situation wherein G is strictly decreasing follows upon observing that G − is also strictlydecreasing and that consequently G ′ ◦ G − ( x ) < holds. Hence, concavity of χ on the set (0 , G ( t ∗ )] ,implies ψ ′ ( t ) ≥ as was to be shown.When applying the previous result with the family { G α } α< of (4), we obtain the followingcorollary. Corollary 9 (Characterisation in the case of G α ) . Let α ∈ ( −∞ , be given and F : R → [0 , bethe cumulative distribution function of a random variable with continuously differentiable density f .Then we have the following equivalence1. f is G α -decreasing (i.e., f is (1 − α ) -decreasing in the sense of Definition 2.2 in [24])2. F is concave- G α (i.e., F is α -revealed-concave in the sense of Definition 3.1 in [65]). The implication ⇒ of Corollary 9 for α < was already known and corresponds to Lemma3.1 in [24]. However both the extension to α ∈ [0 , and the reverse implication are novel. Especiallythe latter shows that, in principle, there is no loss of generality in studying the properties of thedensity instead of the cumulative distribution function F .As already mentioned, [24, Table 1] contains a large choice of usual distribution functions (nor-mal, exponential, Weibull, gamma, chi, Maxwell, etc...) with a (1 − α ) -decreasing density function11or all α < and an analytic expression for the parameter t ∗ indicated in Definition 7. Althoughthese results may give the impression that all cumulative distribution functions are concave − G α − ,this is not true as the following example shows. Example − G α − distribution function) . Let f : R + → R be defined as f ( t ) =( sin ( t ) t ) π . Let us first verify that f does indeed integrate to . This follows recalling the iden-tity sin ( t ) = (1 − cos(2 t )) / and by using integration by parts, as well as by recalling that lim t →∞ sin( s ) s ds = π .Now, should f be G α -decreasing for some α < , then it must hold by (9) that − α ( α − α t − α f ( t ) + 1 α t − α f ′ ( t ) ≤ , for t ≥ t ∗ for some t ∗ . Yet the previous inequality is equivalent with (1 − α ) f ( t ) + tf ′ ( t ) ≥ for t ≥ t ∗ .Let us verify that this can not hold. Note that f ′ ( t ) = (cid:18) − ( t ) t + 2 sin( t ) cos( t ) t (cid:19) π . We verify the negativeness of expression g ( t ) := (1 − α ) f ( t ) + tf ′ ( t ) , after algebraic manipulations: g ( t ) = 2 π sin( t ) t (cid:18) ( − − α ) sin( t ) t + 2 cos( t ) (cid:19) Case − < α < : Choosing the points t n = π +2 πn , for integers n > , we note that cos( t n ) = 0 , sin( t n ) t n > and ( − − α ) sin( t n ) t n < . Hence, there is always t n such that g ( t n ) < .Case α ≤ − : Choosing the points t n = π + 2 πn , for integers n > sufficient large such that t n > − − α . By noting that cos( t n ) = − . and sin( t n ) = 0 . , it is easy to verify that ( − − α ) sin( t n ) t n + 2 cos( t n ) < . Again, consequently, there is always t n such that g ( t n ) < .Would the requested t ∗ exist, we must have for some n sufficiently large that t n > t ∗ andconsequently condition (9) must hold in particular at t = t n . Yet, we have established that it cannot. Hence the density function is not G α -decreasing for any α < . Example G − with G = G α ) . Let us come back to Example 2 and the map G : R + → R + defined as G ( x ) = exp( − ln( x ) ) . Then, Φ the cumulative distribution function of a standard normalGaussian random variable is concave- G − on the set (0 , G ( t ∗ )] , with t ∗ = 1 . .This follows from Proposition 8 as soon as the ratio r ( x ) = √ πG ( x ) exp( − x )( − x ln( x ) ) isstrictly increasing. This, in turn, can be asserted if the function f ( x ) := exp( − x )( − x ln( x ) ) is strictly increasing on the set x ≥ t ∗ . In order to show this, compute the derivative: f ′ ( x ) =3 exp( − x ) ln( x ) (( x −
1) ln( x ) − ) . Observing that for x ∈ (1 , ∞ ) , (3 exp( − x ) ln( x ) ) > ,the sign of f ′ ( x ) depends on the term: f ( x ) = ( x −
1) ln( x ) − . The latter has derivative: f ′ ( x ) = 2 x ln( x ) + ( x − x . For x > , f ′ ( x ) > ( f ( x ) is strictly increasing for x > ). We find that f (1 .
8) = − . and f (1 .
9) = 0 . , so there is a root of f in the interval [1 . , . . So, for x ∈ [1 . , ∞ ) , f ( x ) > and hence f ′ ( x ) > , which in turn implies that f ( x ) is strictly increasing.Numerically solving f ( x ) = 0 in x , we find the solution x ∗ = 1 . .12 Interplay of generalized concavity and “convexity" of chance con-straints
In this section, we establish convexity results for feasible sets of probabilistic constraints by em-ploying the set of tools of generalized concavity. Our analysis considers two special structures for(1). The first situation, analyzed in Section 3.1, refers to the general case wherein g is non-linearand relatively arbitrary, but the random vector ξ is assumed to follow a multi-variate ellipticallysymmetric distribution. The second situation, analyzed in Section 3.2, is when g is separable, whichboils down to considering g ( x, z ) = z − h ( x ) . The random vector ξ can be relatively arbitrary in somuch that it can have nearly arbitrary marginal distributions and the (joint) dependency structureis pinned down by the choice of a copula. In this section we consider the situation of (1) wherein the map g : X × R m → R k is convex in thefirst argument and continuous as a function of both arguments. We also assume that the randomvector ξ taking values in R m is elliptically symmetrically distributed with mean µ , covariance-likematrix Σ and generator θ : R + → R + , which is denoted by ξ ∼ E ( µ, Σ , θ ) if and only if its density f ξ : R m → R + is given by f ξ ( z ) = (cid:0) det Σ (cid:1) − / θ (cid:18) ( z − µ ) ⊤ Σ − ( z − µ ) (cid:19) , (10)where the generator function θ : R + → R + must satisfy Z ∞ t m θ ( t ) dt < ∞ . We consider L as the matrix arising from the Choleski decomposition of Σ , i.e., Σ = LL T , it can beshown that ξ admits a representation as ξ = µ + R Lζ (11)where ζ has a uniform distribution over the Euclidean m -dimensional unit sphere S m − := { z ∈ R m : P mi =1 z i = 1 } and R possesses a density, which is given by f R ( r ) := 2 π m Γ( m ) r m − θ ( r ) , (12)with Γ is the usual gamma-function.The family of elliptically distributed random vectors includes many classical families (see e.g. [17]and [35]): for instance, Gaussian random vectors and Student random vectors (with ν degrees offreedom) are elliptical with the respective generators θ Gauss ( t ) = exp( − t/ / (2 π ) m/ and θ Student ( t ) = Γ (cid:0) m + ν (cid:1) Γ (cid:0) ν (cid:1) ( πν ) − m/ (cid:0) tν (cid:1) − m + ν . A general separable g would be of the form g ( x, z ) = ψ ( z ) − h ( x ) . In our case, recalling that z will be substitutedout for the random vector ξ , there is no loss of generality in assuming ψ ( z ) = z since the general case reduces to itwhen taking ˜ ξ = ψ ( ξ ) to be the underlying random vector that we study. ϕ (see e.g. Theorem 2.1 of [65]): if x ∈ X is such that1. g ( x, µ ) ≤ (recall that µ = E ( ξ ) )2. for any z ∈ R m such that g ( x, z ) ≤ , we have g ( x, λµ + (1 − λ ) z ) ≤ ∀ λ ∈ [0 , , then ϕ defined in (1) can be written as ϕ ( x ) = Z v ∈ S m − F R ( ρ ( x, v )) dµ ζ ( v ) (13)where F R is the cumulative distribution function of R , µ ζ is the law of uniform distribution on the m -dimensional euclidian sphere S m − , and ρ : X × S m − → R + ∪ {∞} is the continuous mappingdefined by ρ ( x, v ) = sup t ≥ ts.t. g ( x, µ + tLv ) ≤ . (14)Note that if for each v the map ρ ( · , v ) is F R -concave, then due to the linearity of the integralsuch a property would carry over immediately to ϕ . It is clear that such a request could not holdwithout restrictions since generally a probability function can not be “concave”. Indeed, it is abounded function by and and usually increasing along a certain “path”. For this reason theanalysis is non-trivial. The second difficulty is in conveying these desired properties from p alone.Let us provide a precise statement. Theorem 10 (Convexity of probability functions) . Let C be a convex subset of X . Assume that,for any v ∈ S m − , there exists a continuous function G v : ρ ( C × { v } ) → R + such that • G v is strictly monotonic on ρ ( C × { v } ) , • x ρ ( x, v ) is G v -concave on C and continuous, • F R is concave − G v − on ( G v ◦ ρ )( C × { v } ) ,where ρ is defined as in (14) . Then ϕ : X → [0 , defined in (1) is concave on C .Proof. Let us first establish that for any fixed v ∈ S m − that x F R ◦ ρ ( x, v ) is concave on C .To this end pick x , x ∈ C and λ ∈ [0 , arbitrarily and consider x λ = λx + (1 − λ ) x . Then by G v -concavity of ρ ( · , v ) it follows: ρ ( x λ , v ) ≥ G v − ( λG v ( ρ ( x , v )) + (1 − λ ) G v ( ρ ( x , v ))) . Now since F R is increasing as a distribution function it follows too that F R ( ρ ( x λ , v )) ≥ F R ( G v − ( λG v ( ρ ( x , v )) + (1 − λ ) G v ( ρ ( x , v )))) . The continuity of ρ is well-known (see e.g. Lemma 3.4 in [64]); it implies in particular that ρ is mesurable. Notealso that ρ is quasi-concave (see Lemma 3.2 in [65]). We will need here stronger notions of G -concavity to establishour results. I v ⊆ R defined as I v = G v ( ρ ( C, v )) .Our claim is that I v is an interval, i.e., is convex. To this end, we recall that the continuousimage of a connected set is connected and hence ρ ( C, v ) is a connected set (recall that C is a convexset). By applying the argument a second time, since G v is continuous, it follows that I v is connected.Now since I v is a subset of R , it is connected if and only if it is convex if and only if it is an interval.Now since I v is an interval and hence convex, λG v ( ρ ( x , v )) + (1 − λ ) G v ( ρ ( x , v ))) ∈ I v and wemay apply concavity − G v − of F R to pursue our development as follows: F R ( ρ ( x λ , v )) ≥ ( F R ◦ G v − )( λG v ( ρ ( x , v ))+(1 − λ ) G v ( ρ ( x , v )))) ≥ λF R ( ρ ( x , v ))+(1 − λ ) F R ( ρ ( x , v )) , which is what was to be shown. Now by linearity of integrals, we get ϕ ( x λ ) = Z v ∈ S m − F R ( ρ ( x λ , v )) dµ ζ ( v ) ≥ λ Z v ∈ S m − F R ( ρ ( x , v )) dµ ζ ( v ) + (1 − λ ) Z v ∈ S m − F R ( ρ ( x , v )) dµ ζ ( v )= λϕ ( x ) + (1 − λ ) ϕ ( x ) , thus concluding the proof.Although Theorem 10 allows us to establish concavity of ϕ on a certain given convex set C ,an important additional difficulty is how to entail that M ( p ) ⊆ C holds for p large enough. Thenone can immediately deduce the convexity of M ( p ) from concavity of ϕ . A convenient situationis one when, for all x , the set M ( x ) := { z ∈ R m : g ( x, z ) ≤ } is convex in R m . For this itwould be sufficient to request that g is convex respectively in x and in z (but not necessarilyjointly). We can however generalize to the situation wherein M ( x ) is star-shaped with respect to µ if the convex hull of the latter sets does “not distort" measurement of length of lines segments { r ≥ µ + rLv ∈ M ( x ) } moving through it. In order to make a precise statement, we introducethe following map: ρ co : X × S m − → R + ∪ {−∞ , ∞} : ρ co ( x, v ) = sup t ≥ ts.t. µ + tLv ∈ Co( M ( x )) (15)where Co( M ( x )) denotes the convex hull of M ( x ) . Theorem 11 (Eventual convexity of elliptical chance constraints) . Let C be a given convex subsetof X . In addition to the framework of this section, assume that1. there exists a t ∗ > , such that { x ∈ X : ρ ( x, v ) ≥ t ∗ , ∀ v ∈ S m − } ⊆ C ;2. For any v ∈ S m − , there exists a continuous function G v : R → R as in Theorem 10;3. There exists p ∈ [ , and δ nd > such that δ nd ρ ( x, v ) ≥ ρ co ( x, v ) , for all x ∈ M ( p ) and all v ∈ S m − , where ρ and ρ co are defined respectively in (14) and (15) .Then for any q ∈ (0 , ) and any p ≥ max( p , p ( t ∗ , q )) with p ( t ∗ , q ) = (cid:18) − q (cid:19) F R (cid:18) δ nd t ∗ δ ( q ) (cid:19) + 12 + q, (16)15 he set M ( p ) defined in (2) is convex. Here δ ( q ) is the unique solution (in δ ) to the equation B i (cid:18) m − , , sin (arccos( δ )) (cid:19) = (1 − q ) B c (cid:18) m − , (cid:19) , where B i (resp. B c ) refers to the incomplete (resp. complete) Beta function.Proof. We follow here closely the demonstration of the Theorem 4.1 from [65]. Let p ∈ [ p , begiven and take any x ∈ M ( p ) . We have, for such x that < p ≤ P [ g ( x, ξ ) ≤ ≤ P [ ξ ∈ Co ( M ( x ))] .Then corollary 2.1 from [65] gives that µ ∈ int ( Co ( M ( x ))) .Let us now pick an arbitrary but fixed v ∈ D om( ρ ( x, . )) . Note that, by assumption, there is δ nd > such that δ nd ρ ( x, v ) ≥ ρ co ( x, v ) , i.e., v ∈ D om( ρ co ( x, . )) as well. Hence, µ + ρ co ( x, v ) Lv belongs to the boundary of Co( M ( x )) .Therefore, we can separate µ + ρ co ( x, v ) Lv from the convex set Co( M ( x )) , so that there existsa non-zero s ∈ R m such that for all z ∈ Co( M ( x )) , s T z ≤ s T ( µ + ρ co ( x, v ) Lv ) ≤ s T ( µ + δ nd ρ ( x, v ) Lv ) . Now define c ∈ R m and γ > as follows: c := s k L T s k and γ = c T ( µ + δ nd ρ ( x, v ) Lv ) , where we recall the (cid:13)(cid:13) L T s (cid:13)(cid:13) > since L is regular and s = 0 . It now follows by construction that, M ( x ) ⊂ Co( M ( x )) ⊂ { z ∈ R m : c T z ≤ γ } . In particular this entails P [ g ( x, ξ ) ≤ ≤ P [ c T ξ ≤ γ ] . We can employ Theorem 2.2 of [65] to getthe estimate p ≤ P [ g ( x, ξ ) ≤ ≤ P [ c T ξ ≤ γ ] ≤ (cid:18) − q (cid:19) F R δ nd ρ ( x, v ) s T Lv k L T s k δ ( q ) + q + 12 ≤ (cid:18) − q (cid:19) F R (cid:18) δ nd ρ ( x, v ) δ ( q ) (cid:19) + q + 12 , for any q ∈ (0 , ) and associated δ ( q ) > , where we have used the Cauchy-Schwartz inequality andthe monotonicity of F R . Since F R is increasing and ( − q ) > as well as p ≥ max( p , p ( t ∗ , q )) , wederive ρ ( x, v ) ≥ t ∗ . Moreover obviously, ρ ( x, v ) ≥ t ∗ for v / ∈ D om( ρ ( x, . )) . Hence, M ( p ) ⊆ { x ∈ X : ρ ( x, v ) ≥ t ∗ , ∀ v ∈ S m − } ⊆ C . Now, we can apply Theorem 10 to establish that ϕ is concaveon C and therefore M ( p ) must be convex. Remark . Though looking abstract, the conditions of the theorem areoften present in practice. They can for example be ensured whenever the following conditions hold.We provide examples in Section 4.• There exist some α ∈ R such that for each v ∈ S m − , the map x ρ ( x, v ) is α -concave (see[65, Proposition 5.1] for an exemple) and the radial distribution function F R is concave- α .Here prominent examples are the chi distribution, the Fisher-Snedecor distribution, etc.16 To get a suitable t ∗ associated with F R , one can use the concavity- α of F R . For the example ofthe chi-distribution, we get t ∗ = √ m − α and C = (cid:8) x ∈ X : ρ ( x, v ) ≥ √ m − α for all v ∈ S m − (cid:9) .• The request of item 3 holds whenever the set M ( x ) is convex for all x ∈ C and in that case δ nd = 1 . Such convexity can be ensured whenever the map z g ( x, z ) is quasi-convex foreach x ∈ X , and in that case p = can be taken.• Note finally that continuity of the map ρ can be ensured under fairly general conditions, e.g.,convexity of g in the second argument and g ( x, µ ) < together with continuous differentiabilityof g immediately entail continuity of x ρ ( x, v ) (even continuity in both arguments). See forinstance [21]. Continuity can also be ensured under less restrictive conditions. For instancewhenever, the map g is continuous at any ¯ x, ¯ v, ¯ r such that g (¯ x, ¯ rL ¯ v ) = 0 , neighbourhoods U, V, W of ¯ x, ¯ v, ¯ r respectively can be identified such that for all ( x, v ) ∈ U × V , the map r g ( x, rLv ) is monotonic. This in turn is related to uniqueness of solutions of perturbedsystems g ( x, rLv ) = t for t sufficiently small and not very restrictive. To ensure continuity of ρ from this assumption, we can just use a general version of the implicit function theorem; see[13, Theorem 1H.3] as well as [27, 33] which are older statements of such a result.The computed threshold p ∗ is thus valid for a large class of nonlinear functions g . Should it beconservative, refinements might be obtained when studying specific structures; see e.g., [43, 57] andforthcoming Remark 3. Beyond being a general guarantee, the threshold p ∗ is thus an indicationthat lower thresholds could be revealed for specific functions. In this section we will consider the following form of (1): P [ ξ ≤ h ( x )] ≥ p, where ξ taking values in R m is a random vector and h : X → R m a given map. By employingSklar’s Theorem [52], we may write (1) with the special structure as above in the following form: P [ ξ ≤ h ( x )] = C ( F ( h ( x )) , ..., F m ( h m ( x ))) . (17)Here F , ..., F m are the marginal distribution functions of the random vector ξ and h i : X → R , i = 1 , ..., m refers to the components of the mapping h . Moreover C : [0 , m → [0 , is a copula,i.e., a multi-variate distribution function with uniform marginal distributions (e.g., [44] for furtherdetails). In what follows, we require specific properties of C and F , ..., F m and through these choicespin down the multivariate distribution of ξ .We introduce the analogue of concave- G − functions in this context of copulæ. Definition 12 (concave- G − copulæ) . Let C : [0 , m → [0 , be a copula and G : [0 , m → R m amap such that the i th component G i is continuous and strictly monotonic. The copula C is said tobe concave- G − on the product of intervals I = Q mi =1 I i , with I i ⊆ R if the map I ∋ z
7→ C ( G − ( z ) , ..., G m − ( z m )) is quasi-concave. 17 xample . Taking G i = G γ in the previous definition, aconcave- G − copula corresponds to a ( −∞ ) − γ -concave copula in the terminology of [56]. Forexample, all Archimedian copulæ are concave − G − (see [59, Theorem 3.3]. Example 7 belowprovides an example of concave- G − copula which is necessarily ( −∞ ) − γ -concave. Note also that,in the special case γ = 0 , this notion is weaker than the notion of logexp-concave of [25]). Prominentexamples of such copulæ are for instance the independent, maximum or Gumbel copula. The Claytoncopula is an example of concave − G − copula which is not logexp-concave (see Lemma 5.5 of [56]). Example . Let R be a positive definite m × m correlation matrix and Φ denote thestandard normal distribution function. Then the Gaussian copula C R : [0 , m → [0 , is defined as C R ( u ) = Φ R ( Φ − ( u ) , ..., Φ − ( u m )) , where Φ R is the multivariate Gaussian distribution function related to correlation matrix R . Nowobserve that Φ − : [0 , → R is strictly increasing and continuous. Now for any z ∈ I = R m , C R (Φ( z ) , ..., Φ( z m )) = Φ R ( Φ − (Φ( z )) , ..., Φ − (Φ( z m ))) = Φ R ( z , ..., z m ) . Recalling that multivari-ate Gaussian distribution functions are -concave (see, e.g., [47]), it follows from Corollary 4 that C R is concave − Φ − − . It was however not clear whether or not C R is ( −∞ ) − γ -concave; see theextensive discussion in [56, section 6]). Similar analysis can be carried out with Clayton Copulawhich can be shown to be concave- G − with G i ( z ) = ln ( z ) .We can now provide the announced eventual convexity result. Theorem 13 (Eventual convexity of separable copulae-structured probabilistic constraints) . Let h i : X → R be continuous mappings and consider the following identity: P [ ξ ≤ h ( x )] = C ( F ( h ( x )) , ..., F m ( h m ( x ))) , (18) where C is a suitable copula and F i are the marginal distribution functions of component i of ξ , i = 1 , ..., m .Assume that we can find strictly monotonous mappings G i : R → R , such that the functions h i are G i -concave on a given convex level set C = { x ∈ X : h i ( x ) ≥ b i , i = 1 , ..., m } of X for appropriateparameters b i ∈ R , i = 1 , ..., m .Assume moreover that for i = 1 , ..., m continuous strictly monotonic mappings ˆ G i : [0 , → R can be identified such that C is concave − ˆ G − (see Definition 12) on the set I = Q mi =1 I i , where • the interval I i = [ ˆ G i ( b i ) , ∞ ) whenever ˆ G i is strictly increasing • the interval I i = ( −∞ , ˆ G i ( b i )] whenever ˆ G i is strictly decreasing.Finally, assume that the marginal distribution functions F i : R → R are ˆ G i -concave- G − i on • the interval [ G i ( b i ) , ∞ ) whenever G i is strictly increasing • the interval (0 , G i ( b i )] whenever G i is strictly decreasing.Then the set M ( p ) := { x ∈ X : P [ ξ ≤ h ( x )] ≥ p } is convex for all p > p ∗ := max i =1 ,...,m F i ( b i ) .Convexity can also be asserted for p = p ∗ if each individual distribution function F i , i = 1 , ..., m isstrictly increasing. roof. Pick any p > p ∗ , x, y ∈ M ( p ) , λ ∈ [0 , and i ∈ { , ..., m } arbitrarily. Define x λ := λx + (1 − λ ) y . Since all copulæ are dominated by the maximum-copula, we get: F i ( h i ( x )) ≥ min j =1 ,...,m F j ( h j ( x )) ≥ C ( F ( h ( x )) , ..., F m ( h m ( x ))) ≥ p > p ∗ ≥ F i ( b i ) . (19)Now the latter entails h i ( x ) ≥ b i . (20)and this in turn means M ( p ) ⊆ C . Estimate (20) also holds whenever p ≥ p ∗ and F i is strictlyincreasing for each i = 1 , ..., m . A similar estimate is obtained for y clearly. Let us first remark thatdue to (20), we have G i ( h i ( x )) ≥ G i ( b i ) , whenever G i is strictly increasing and G i ( h i ( x )) ≤ G i ( b i ) otherwise.Consequently, λG i ( h i ( x )) + (1 − λ ) G i ( h i ( y )) ≤ G i ( b i ) whenever G i is strictly decreasing, and λG i ( h i ( x )) + (1 − λ ) G i ( h i ( y )) ≥ G i ( b i ) otherwise. We note that in either situation λG i ( h i ( x )) +(1 − λ ) G i ( h i ( y )) belongs to the interval associated with ˆ G i -concavity- G − i of F i . Now, since F i isincreasing we establish first that F i ( h i ( x λ )) ≥ F i ( G i − ( λG i ( h i ( x )) + (1 − λ ) G i ( h i ( y )))) , (21)by G i -concavity of h i on the set C . As argued above, related to λG i ( h i ( x )) + (1 − λ ) G i ( h i ( y )) belonging to an appropriate domain, we may pursue (21) by invoking ˆ G i -concavity- G − i of F i , andobtain F i ( h i ( x λ )) ≥ F i ( G i − ( λG i ( h i ( x )) + (1 − λ ) G i ( h i ( y )))) ≥ ˆG − i ( λ ˆ G i ( F i ( G i − ( G i ( h i ( x ))))) + (1 − λ ) ˆ G i ( F i ( G i − ( G i ( h i ( x )))))) ≥ ˆG − i ( λ ˆ G i ( F i ( h i ( x ))) + (1 − λ ) ˆ G i ( F i ( h i ( y )))) (22)Since i was fixed but arbitrary, the above equation (22) holds for all i = 1 , ..., m . A Copula isincreasing in its arguments, so we get in turn: C ( F ( h ( x λ )) , ..., F m ( h m ( x λ ))) ≥C ( ˆG − ( λ ˆ G ( F ( h ( x ))) + (1 − λ ) ˆ G ( F ( h ( y )))) , ..., ˆG − m ( λ ˆ G m ( F m ( h m ( x ))) + (1 − λ ) ˆ G m ( F m ( h m ( y ))))) . Moreover arguing as before, for any i = 1 , ..., m it holds that λ ˆ G i ( F i ( h i ( x )))+(1 − λ ) ˆ G i ( F i ( h i ( y ))) ∈ I i . We can now employ concavity − ˆ G − of the copula C , to establish: C ( F ( h ( x λ )) , ..., F m ( h m ( x λ ))) ≥ m −∞ ( C ( F ( h ( x )) , ..., F m ( h m ( x ))) , C ( F ( h ( y )) , ..., F m ( h m ( y ))) , λ ) ≥ p, which is equivalent with C ( F ( h ( x λ )) , ..., F m ( h m ( x λ ))) ≥ p , i.e., x λ ∈ M ( p ) as was to be shown.The conditions given in the above Theorem 13 allow for quite some flexibility. We refer to [56,section 5] for examples. In section 4.2 below we provide other examples, not covered by prior results. In this section, we provide several new examples for which eventual convexity can be asserted withthe extended framework built upon G -concavity. Section 4.1 is concerned with a situation wherein g is non-linear and Section 4.2 wherein g separable.19 .1 Example with a quadratic probabilistic constraint Consider the map g : R n × R m → R defined as g ( x, z ) = z T W ( x ) z + 2 n X i =1 a i w T i z + b (23)where a , ..., a n are fixed constants, w , ..., w n are fixed vectors of R m , and W : R n → S + n is a “convex"function, in the sense that for any ( x, y ) ∈ R n , λ ∈ [0,1], λW ( x ) + (1 − λ ) W ( y ) − W ( λx + (1 − λ ) y ) is positive (semi-)definite. Note that, for all z , g ( · , z ) is then convex with respect to its first variable.We assume furthermore that b < , which ensures g ( x, < . A concrete example of a mapping W satisfying the above request is for instance W ( x ) = P ni =1 x i W i , with W i positive semi-definiteand x ≥ as an ambiant further restriction.Let ξ be an elliptically symmetrically distributed random vector with given spherical-radialdecomposition ξ = R Lζ . Then, by fixing ( x, v ) ∈ R n × S m − and defining h ( x, v ) := ( Lv ) T W ( x )( Lv ) and β ( v ) := n X i =1 a i w T i ( Lv ) , we can explicitly identify the map ρ of (13), being the unique solution to the equation g ( x, rLv ) = 0 .This solution map is given by the expression: ρ ( x, v ) = − β ( v ) + p β ( v ) − bh ( x, v ) h ( x, v ) . (24)Next, we identify a function g v : R + → R such that ρ ( ., v ) is g v -concave for any v ∈ S m − . Lemma 14.
For any fixed v ∈ S m − define the map g v : (0 , ∞ ) → R − as g v ( t ) = − ( bt + β ( v )) .Then x ρ ( x, v ) g v -concave on R n .Proof. We first note that ρ ( x, v ) > , since ρ ( x, v ) is the solution to g ( x, rLv ) = 0 , and z g ( x, z ) is convex (as a convex quadratic map in z ) as well as g ( x, < so that by continuity any solution (ifany) to g ( x, rLv ) = 0 must satisfy r > . Consequently the composition g v ( ρ ( x, v )) is well-defined.Moreover, g v is clearly (continuously) differentiable on (0 , ∞ ) . Now observe that ρ ( x, v ) = h ( x, v ) − β ( v ) + p β ( v ) − bh ( x, v ) = β ( v ) + p β ( v ) − bh ( x, v ) − b . So that, − bρ ( x,v ) − β ( v ) = p β ( v ) − bh ( x, v ) ≥ , i.e., g v ( ρ ( x, v )) = − ( bρ ( x, v ) + β ( v )) = − β ( v ) + bh ( x, v ) . Now, b < and x h ( x, v ) is convex, therefore x g v ( ρ ( x, v )) is concave. It remains to show that g v is strictly increasing on an appropriate domain. To this end, observe that• if β ( v ) ≤ , we have for t > , g ′ v ( t ) = 2 bt ( bt + β ( v )) . Now, since b < and β ( v ) ≤ , it followsthat ( bt + β ( v )) < and evidently bt < so we conclude that g ′ v ( t ) > , as was to be shown.Note that g v maps to ( −∞ , − β ( v ) ) for β ( v ) ≤ .20 if β ( v ) > , we still have, for t > , g ′ v ( t ) = (2 bt ( bt + β ( v )) and bt < . However ( bt + β ( v )) < if and only if t < b − β ( v ) , so that g v is strictly increasing on (0 , b − β ( v ) ] in this case only. We willnow establish that ρ ( x, v ) ∈ (0 , b − β ( v ) ) holds for such v . To this end recall ρ ( x, v ) = β ( v ) + p β ( v ) − bh ( x, v ) − b ⇐⇒ ρ ( x, v ) = − bβ ( v ) + p β ( v ) − bh ( x, v ) , which with β ( v ) + p β ( v ) − bh ( x, v ) > β ( v ) ≥ , gives ρ ( x, v ) < b − β ( v ) as desired.We can identify explicitly the inverse function of g v , but for this we will need to make a case-distinction.• For v such that β ( v ) ≤ , g v − : ( −∞ , − β ( v ) ) → (0 , ∞ ) is given by g v − ( t ) = − b √− t + β ( v ) • For v such that β ( v ) > , g v − : ( −∞ , → (0 , b − β ( v ) ] , is given by g v − ( t ) = − b √− t + β ( v ) .We now employ Lemma 5 in order to combine specialized concavity of ρ with more usual notions. Lemma 15.
Let v ∈ S m − be given and consider the map g v − ( t ) = − b √− t + β ( v ) defined from ( −∞ , − β ( v ) ) to (0 , ∞ ) when β ( v ) ≤ and from ( −∞ , to (0 , b − β ( v ) ] otherwise. This map is − -concave on the set ( −∞ , − β ( v ) ] .Proof. We will show this by a direct computation. In order to do so, let α > − be given butarbitrary and let ψ denote the map t ( g v − ( t ) ) α . The map ψ is clearly twice differentiable onthe appropriate domain and hence establishing the requested generalized concavity of g v − amountsto establishing convexity of ψ . Now we establish ψ ( t ) = ( − b ) α ( √− t + β ( v )) α ψ ′ ( t ) = − ( − b ) α α − t ) − ( √− t + β ( v )) α ψ ′′ ( t ) = ( 1 + α b )( − t ) − ( 1 g v − ( t ) ) α − (cid:0) ( α − √− t − β ( v ) (cid:1) . It is now clear that for β ( v ) ≤ and α > , it holds ψ ′′ ( t ) > , implying the convexity of ψ , i.e., the − − α -concavity of g v − . When β ( v ) > holds, then for α ≥ , we observe that ( α − √− t − β ( v ) ≥ √− t − β ( v ) . The latter is evidently positive whenever t ≤ − β ( v ) . Remark . Let us observe that for any v and x with g ( x, < , g v ( ρ ( x, v )) ∈ ( −∞ , − β ( v ) ] holds. Indeed, let us set − t := − g v ( ρ ( x, v )) = ( bρ ( x,v ) + β ( v )) . Then as observed already: √− t = − bρ ( x,v ) − β ( v ) = p β ( v ) − bh ( x, v ) ≥ . And consequently √− t − β ( v ) = − bρ ( x, v ) − β ( v ) = 1 ρ ( x, v ) ( − b − βρ ( x, v )) . But, ρ ( x, v ) is the unique (positive) solution (in r ) to the equation h ( x, v ) r + 2 β ( v ) r + b = 0 .Therefore we have √− t − β ( v ) = ρ ( x,v ) h ( x, v ) ρ ( x, v ) = h ( x, v ) ρ ( x, v ) ≥ , thus establishing theclaim. 21e can now provide an eventual convexity statement for a probability function involving g definedin (23). Proposition 16.
Consider the probability function ϕ ( x ) := P [ g ( x, ξ ) ≤ , where x ∈ R n is givenand g defined as in (23) . Let ξ taking values in R m be an elliptically symmetrically distributedrandom vector with mean and covariance matrix Σ and associated radial distribution F R . Let F R be concave- ( − on the set (0 , ( t ∗ ) − ] , with t ∗ given by the first assumption of Theorem 11.Then for any q ∈ (0 , ) , the set M ( p ) is convex provided that p ≥ p ∗ with p ∗ := (cid:18) − q (cid:19) F R (cid:18) t ∗ δ ( q ) (cid:19) + 12 + q, where δ ( q ) is as in Theorem 11.Proof. The set C := (cid:8) x ∈ R n : ρ ( x, v ) ≥ t ∗ ∀ v ∈ S m − (cid:9) is a convex set since ρ ( x, v ) is quasi-concavein x as a result of convexity of g in x . Moreover evidently requisite 1. of Theorem 11 holds. Next,let us turn our attention to establishing requisite 2.To this end, let v ∈ S m − be given and let g v : R + → R − be the map defined in Lemma 14.From this Lemma, we know in particular that ρ is g v -concave on the range of ρ ( ., v ) . Hence inparticular on C . Let us thus pick x , x ∈ C , λ ∈ [0 , arbitrarily. We thus obtain the estimate: ρ ( λx + (1 − λ ) x , v ) ≥ g v − ( λg v ( ρ ( x , v )) + (1 − λ ) g v ( ρ ( x , v ))) . Moreover, by Remark 2 g v ( ρ ( x , v )) ∈ ( −∞ , − β ( v ) ] (and likewise for x ) and therefore similarly λg v ( ρ ( x , v )) + (1 − λ ) g v ( ρ ( x , v )) ∈ ( −∞ , − β ( v ) ] . Now, we may apply Lemma 15 to derive thefurther estimate: g v − ( λg v ( ρ ( x , v )) + (1 − λ ) g v ( ρ ( x , v ))) ≥ m − ( g v − ( g v ( ρ ( x , v ))) , g v − ( g v ( ρ ( x , v ))) , λ )= m − ( ρ ( x , v ) , ρ ( x , v ) , λ ) . Finally, since F R is increasing as a distribution function, we may pursue our estimates to obtain: F R ( ρ ( λx + (1 − λ ) x , v )) ≥ F R ( m − ( ρ ( x , v ) , ρ ( x , v ) , λ )) ≥ λF R ( ρ ( x , v )) + (1 − λ ) F R , since z F R ( z − ) is concave on the set (0 , ( t ∗ ) − ] by assumption and ρ ( x , v ) ≥ t ∗ (likewise for x ) so that λρ ( x , v ) − + (1 − λ ) ρ ( x , v ) − ∈ (0 , ( t ∗ ) − ] . This concludes the proof of requisite 2.Moreover note furthermore that the map g defined in (23) is convex in the second argument z . Consequently, for all x ∈ M ( ) , we have g ( x, < and requisite 3. of Theorem 11 holds for p = . The resulting convexity now follows from applying Theorem 11. Remark . When ξ is multi-variate Gaussian, the value t ∗ can be establishedto be √ m + 3 and moreover the threshold can be strengthened to p ∗ = Φ( √ m + 3) . Note moreoverthat, in all cases, we do not only have convexity of M ( p ) for p > p ∗ , but even concavity of ϕ onthis set, which is a rather strong property. Concavity of ϕ is not required to derive convexity of itslevel-sets. Indeed, quasi-concavity would suffice.In view of this, it is of interest to recall results for the specifically structured mapping g ( x, z ) = z − x . In this case, ρ can be proved to be concave [65, Example 3.3]). Using Theorem 11, we would de-rive a stronger property: the concavity of ϕ on the sets of the form (cid:8) x ∈ R m : x ≥ µ + √ m − k L k e (cid:9) (with e the all-one vector); if ξ is Gaussian, this is already known (e.g., [48, Theorem 2.1]). Yet, itis also well-known that multivariate Gaussian distribution functions are log-concave ([34]) and thushave all level sets convex. As a result, the true threshold in that situation would be p ∗ = 0 .22 xample . As a numerical illustration, we propose to take both the decision and random vectorin dimension , i.e., m = n = 2 . We let ξ be a Gaussian random vector with mean µ = 0 and covariance matrix Σ = (cid:18) . . . . (cid:19) . The mapping W : R +2 → S ++2 is given by W ( x , x ) = (cid:18) x + 0 . | x − | + 0 . (cid:19) , a w T + a w T ) = (1 , and b = − .We obtain for these inputs the contour plot of Figure 1. The contour lines are regularly drawnfrom the probability value . to . . We see on this graph how the level sets tend to get convexas the level increases. The red region in the center is the region where ρ ( x, v ) ≥ √ m + 3 = √ forany v ∈ S m − . Proposition 16 allows us to establish convexity for p ∗ = Φ( √
5) = 0 . . We seehere that the set obtained, is far, from being the largest one where convexity seems to be present.This illustrates the discrepancy between the current available threshold p ∗ , depending, in this case,adversely on dimension and the practical situation, which seems to exhibit convexity for thresholdssignificantly lower!We gather in a Python toolbox many useful functions for numerical illustrations in this frame-work. In particular, we provide tools for computing the probability function ϕ (with associatedfunction ρ ) and plotting of its level-sets; as in Figure 1. We call this toolbox p ychance and make itpublicly available on GitHub at https://github.com/yassine-laguel/pychance . Figure 1:
Plot of the probability function ϕ for the given quadratic problem. We provide here eventual convexity statements based on results of Section 3.2 for examples thatare not covered by prior results. We choose these examples with decision vectors of dimension 2to keep calculus simple without misrepresenting our results; this dimension does not impact ourresults, valid in infinite dimension.
Example m -dimensional copula) . Consider any m dimensional Archimedeancopula, and construct the probability functions ϕ ( x ) = C ( F ( h ( x )) , . . . , F m ( h m ( x ))) , with a G -concave h and α -concave h i ( i ≥ ) defined as follows. Given remark 6, we know that C is concave − G − . Let h : R → R ++ be defined by h ( x, y ) = ( yx ) which is neither concave23or convex, but G -concave with G ( x ) = − / √ x as easily shown . Let h i : R → R ++ be definedfrom any convex positive function f i by h i ( x, y ) = f i ( x, y ) α i with α i ∈ [ − , . Let us define nowthe marginal distribution functions. We take F ( z ) = 1 − exp( − λz ) the exponential distributionfunction of parameter λ > ; we easily see that it is concave- G − on the interval I = [ b, ∞ ) with b = p λ/ . We take F i to be the Rayleigh distribution function with parameter σ = 1 . for all i ≥ .We can then apply Theorem 13 to get the convexity of M ( p ) for any p > p ∗ with p ∗ = max (cid:26) F (cid:18) λ (cid:19) , max i ≥ F i (cid:16)p (1 − α i / σ (cid:17)(cid:27) = 0 . since F (cid:0) λ (cid:1) = 0 . and the max over i ≥ does not exceed . . Thus the obtained thresholdis fixed and independent of the dimension of the random vector. Remark . The previous example gives a situation wherein thethreshold p ∗ does not depend on the dimension m of the random vector. The underlying reason isthat all mappings h i are generalized concave with a set of parameters that do not degenerate with m .It should also be observed that the involved parameters (see e.g., [24, Table 1]) depend continuouslyon the parameters of the chosen marginal distributions and the generalized concavity parameter of h . Therefore, should these parameters belong to a compact set, the threshold p ∗ does not dependon dimension either. It is reasonable to assume that the mappings h i , marginal distributions, andtheir parameters are homogenous. Typically the index i expresses time and the probability functionstems from the desire to incorporate akin features: each component i is relatively similar. Example
10 (Convexity statement with functions from the literature) . Let us consider the functionsof Example 1 of [68]: the two mappings from R to R + h ( x, y ) = exp − ( x + y ) and h ( x, y ) = 1 x + y + 1 and the distributions F = Φ , the distribution function of a standard normal Gaussian randomvariable and F = F χ , the distribution function of a χ -random variable with degrees of freedom.From Example 2, the map h can be shown to be G -concave with a given strictly decreasing map G .Moreover, Example 5 gives us that Φ is concave- G − on the set (0 , G (1 . . Observe also that themap h is ( − -concave (indeed, h − is convex) and F χ is concave − ( g − ) − (see Table 1 in [65])on the set (0 , √ − ] .We extend now the situation of Example 1 of [68] which is restricted to the independent copula.We consider here any concave − g − copula (in particular for all Archimedian copulæ, by Example 6),and we apply Theorem 13 to get that the feasible set M ( p ) with probability function ϕ ( x, y ) = C ( F ( h ( x, y )) , F ( h ( x, y ))) , We see that G ( x ) is strictly increasing. To verify that ( x, y ) G ( h ( x, y )) = − x y is concave, just compute itsHessian − y (cid:18) y − xy − xy x (cid:19) which is is negative semi-definite by a direct computation (negative trace and zero determinant). Observe first that G − ( z ) = z − is convex and strictly increasing. Verify that the composition F ( G − ( z )) isconcave on some subset when z takes arguments in ( −∞ , , by a direct computation of its second derivative whichgives f ( G − ( z )) z − (3 − λz − ) with the density function f ( z ) = λ exp( − λz ) .
24s convex for p ∗ = max (cid:8) Φ(1 . , F χ ( √ (cid:9) = max { . , . } = 0 . . Existing theory cannotbe applied to this case: [68] requires independent copula, and [56] requires α -concavity (but h isnot α -concave for any α ). Example
11 (Improved previous result for specific copula) . Let us go further with the previousexample when C is concave − G − which subsumes the case of independent copula considered in[68]. In this case, z Φ( G − ( z )) is log-concave on the larger set (0 , G ( t z )] , with t z = 1 . (thisvalue is numerically identified by employing the principles of [56, section 5.2]), and z F χ ( z − ) islog-concave on an interval of the form (0 , . (by Corollary 9). Then for any concave − G − copula (e.g., independent, Gumbel, Clayton), Theorem 13 gives the threshold p ∗ = max n Φ( t z ) , F χ (1 . o = max { Φ(1 . , . } = 0 . . This value is better than the threshold
Φ(3) = 0 . given in [68, Example 1] for the independentcase only. Concluding Remarks
In this paper we have provided general conditions under which probabilistic constraints define aconvex set. We have investigated two different structures of the probability functions: (i) a non-separable structure with elliptically random vectors; (ii) a separable structure with the dependencyexpressed by a given copula. In these situations, we have employed the notion of G -concavity toreveal the level of underlying convexity in the functions defining the constraints. The obtainedresults are more general than those appearing in prior results. The provided conditions can beverified from the nominal problem data, as illustrated on various examples.This work raises issues about the application of our convexity results in practice for generalchance-constrained optimization problems (non-emptiness and convexity of feasible sets, guaranteesof solutions... see the discussion in Section 1.3). A related question is the improvement of thecomputed thresholds. In some specific situations indeed, it might be possible to lower the computedthreshold p ∗ by exploiting structure of the problems. More generally, more work is required in orderto reduce to gap between observed convexity and guaranteed convexity. Acknowledgments
We would like to acknowledge the partial financial support of PGMO (Gaspard Monge Programfor Optimization and operations research) of the Hadamard Mathematic Foundation, through theproject “Advanced nonsmooth optimization methods for stochastic programming".
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