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Convexity of chance constraints for elliptical and skewed distributions with copula structures dependent on decision variables

Heng Zhang, Abdel Lisser

TL;DR

This paper tackles the convexity of joint chance constraints in settings where rows of the random constraint matrix are elliptically distributed and linked through a copula that depends on the decision vector. By decomposing the joint constraint into scalar components and a copula-driven aggregation, it derives best $r$-concavity thresholds and establishes eventual convexity for elliptical distributions, even in the presence of a singular copula at the origin. The work extends these results to skewed distributions using generalized hyperbolic (GH) and normal mean-variance mixture (NMVM) models, proving $\ extalpha$-decreasing density properties and employing radial decompositions to show eventual convexity under broader conditions. Theoretical contributions are complemented by simulations that illustrate the existence of thresholds and the practical impact on feasible set convexity, with implications for robust optimization under uncertainty. Overall, the results unify and extend convexity guarantees for chance-constrained problems with complex dependence structures and skewness, enabling tractable reformulations and solution methods in engineering and finance applications.

Abstract

Chance constraints describe a set of given random inequalities depending on the decision vector satisfied with a large enough probability. They are widely used in decision making under uncertain data in many engineering problems. This paper aims to derive the convexity of chance constraints with row dependent elliptical and skewed random variables via a copula depending on decision vectors. We obtain best thresholds of the $r$-concavity for any real number $r$ and improve probability thresholds of the eventual convexity. We prove the eventual convexity with elliptical distributions and a Gumbel-Hougaard copula despite the copula's singularity near the origin. We determine the $α$-decreasing densities of generalized hyperbolic distributions by estimating the modified Bessel functions. By applying the $α$-decreasing property and a radial decomposition, we achieve the eventual convexity for three types of skewed distributions. Finally, we provide an example to illustrate the eventual convexity of a feasible set containing the origin.

Convexity of chance constraints for elliptical and skewed distributions with copula structures dependent on decision variables

TL;DR

This paper tackles the convexity of joint chance constraints in settings where rows of the random constraint matrix are elliptically distributed and linked through a copula that depends on the decision vector. By decomposing the joint constraint into scalar components and a copula-driven aggregation, it derives best -concavity thresholds and establishes eventual convexity for elliptical distributions, even in the presence of a singular copula at the origin. The work extends these results to skewed distributions using generalized hyperbolic (GH) and normal mean-variance mixture (NMVM) models, proving -decreasing density properties and employing radial decompositions to show eventual convexity under broader conditions. Theoretical contributions are complemented by simulations that illustrate the existence of thresholds and the practical impact on feasible set convexity, with implications for robust optimization under uncertainty. Overall, the results unify and extend convexity guarantees for chance-constrained problems with complex dependence structures and skewness, enabling tractable reformulations and solution methods in engineering and finance applications.

Abstract

Chance constraints describe a set of given random inequalities depending on the decision vector satisfied with a large enough probability. They are widely used in decision making under uncertain data in many engineering problems. This paper aims to derive the convexity of chance constraints with row dependent elliptical and skewed random variables via a copula depending on decision vectors. We obtain best thresholds of the -concavity for any real number and improve probability thresholds of the eventual convexity. We prove the eventual convexity with elliptical distributions and a Gumbel-Hougaard copula despite the copula's singularity near the origin. We determine the -decreasing densities of generalized hyperbolic distributions by estimating the modified Bessel functions. By applying the -decreasing property and a radial decomposition, we achieve the eventual convexity for three types of skewed distributions. Finally, we provide an example to illustrate the eventual convexity of a feasible set containing the origin.

Paper Structure

This paper contains 23 sections, 31 theorems, 308 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Eventual convexity with elliptical distributions and copulas
  3. Preliminaries
  4. Reformulation of chance constraints
  5. Concavity of $H_{i}(x)$
  6. Convexity of $U(x, y_{i})$
  7. Convexity of $S(p)$ under elliptical distributions
  8. Extension of eventual convexity under skewed distributions
  9. Fundamental about skewed distributions
  10. Convexity under less skewed GH distributions
  11. Convexity with NMVM distributions
  12. Convexity under strong skewed GH distributions
  13. Simulations about eventual convexity with elliptical distributions
  14. Numerical simulation of eventual convexity.
  15. Examples about the threhold $\sqrt{\theta^{*}}$.
  16. ...and 8 more sections

Key Result

Proposition 1

(Frechet-Hoeffding upper bound)nelsen2006introduction For each copula $C$ and $u = (u_{1}, ..., u_{K})$, we have

Figures (5)

  • Figure 1: The surface of a probability function with Student distribution.
  • Figure 2: The contour line of a probability function with Student distribution.
  • Figure 3: The graphs about non-existence of $\sqrt{\theta^{*}}$.
  • Figure 4: The graphs about existence of $\sqrt{\theta^{*}}$.
  • Figure 5: The graph of $\sqrt{\theta^{*}}$ with $b > 0$

Theorems & Definitions (73)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Definition 4
  • Lemma 3
  • Remark 1
  • Lemma 4
  • Lemma 5
  • ...and 63 more